AHEAD : Ad-Hoc Electronic Auction Design
Joffrey Derchu, Philippe Guillot, Thibaut Mastrolia, Mathieu Rosenbaum
AAHEAD :
Ad-Hoc
Electronic Auction Design
Joffrey Derchu ∗ , Philippe Guillot † , Thibaut Mastrolia ‡ and Mathieu Rosenbaum § October 7, 2020
Abstract
We introduce a new matching design for financial transactions in an electronic market. Inthis mechanism, called ad-hoc electronic auction design (AHEAD), market participants cantrade between themselves at a fixed price and trigger an auction when they are no longersatisfied with this fixed price. In this context, we prove that a Nash equilibrium is obtainedbetween market participants. Furthermore, we are able to assess quantitatively the relevanceof ad-hoc auctions and to compare them with periodic auctions and continuous limit orderbooks. We show that from the investors’ viewpoint, the microstructure of the asset is usuallysignificantly improved when using AHEAD.
Keywords:
Market microstructure, market design, financial regulation, ad-hoc auctions, periodicauctions, limit order book, Nash equilibrium.
The question of a suitable market microstructure enabling an exchange to ensure satisfactory con-ditions for trading activities of market participants is particularly intricate. The most standardapproach, adopted by a large number of exchanges, is the continuous limit order book (CLOB forshort). In this setting, market participants can either choose to trade immediately by acceptingthe price offered by a counterparty in the order book (sending what is called an aggressive order ∗ ´Ecole Polytechnique, CMAP; joff[email protected] † Autorit´e des March´es Financiers; [email protected] ‡ ´Ecole Polytechnique, CMAP; [email protected] § ´Ecole Polytechnique, CMAP; [email protected] a r X i v : . [ q -f i n . T R ] O c t nd thereby reducing the quantity of shares instantly available in the limit order book) or place a passive order, which waits in the order book to find a counterparty. The recent change in the verynature of market makers, which are nowadays essentially high frequency traders, has triggered adebate on whether CLOBs are the most suitable order matching mechanism, notably in terms ofquality of the price formation process. The alternative design which is usually put forward is thatof periodic auctions. In this case, transactions occur once the auction terminates. The traded priceis the equilibrium price maximising the number of financial instruments traded, determined at theend of the auction period from the imbalance between buy and sell orders accumulated during theduration of the auction. Currently, some auctions are already held at regular intervals in manymarkets where the main mechanism is a CLOB, typically at the beginning and at the end of thetrading day. Moreover, some exchanges organise periodic auctions throughout the day. This is forexample the case of BATS-Cboe for European equities.One of the benefits of auctions derives from the fact that they mechanically slow down the market.Doing so, they suppress some obvious flaws due to speed competition of high frequency market mak-ers in a CLOB environment. This is particularly well emphasized in Farmer and Skouras (2012);Aquilina et al. (2020) and the influential paper Budish et al. (2015) where a lower bound for auctionduration so that speed arbitrages vanish is provided (about 100 ms). In the paper Du and Zhu(2017), the authors also consider the issue of determining a suitable time period for the auctionduration. To do so, they model the behaviour of microscopic agents who optimise their demandschedules with respect to the available information in the market. They show that the optimalauction duration is linked to the rate of arrival of information.Regarding a suitable market design, each type of market participants has a different view on thequestion depending on its activity. This is why there is a crucial need for a quantitative analysisenabling us to assess and compare the different mechanisms objectively. This is done in Jusselin et al.(2019) where the authors extend the works Fricke and Gerig (2018) and Garbade and Silber (1979).More precisely, they are able to compare CLOBs and periodic auctions from a price formationprocess viewpoint using stochastic differential games. They also provide optimal auction durations(a few minutes in practice according to their approach, depending on the asset involved). Auctions and CLOB represent two quite orthogonal approaches in terms of market design. Inthis paper, we aim to study an hybrid mechanism that we call ad-hoc electronic auction design(AHEAD). The idea of ad-hoc auctions is to organise a specific type of continuous trading session2fter each auction. During the continuous session, market participants trade between themselves ata fixed price equal to the last auction’s clearing price. Any market participant has the opportunityto end the continuous session when he is no longer satisfied with the price by triggering a newauction. The only constraint imposed by the exchange to market participants for triggering a newauction phase is to commit at least a minimal volume in the auction. In this setting, there canbe two reasons to motivate investors for ending the continuous phase. Either they consider thetrading price is no longer reasonable or they are not able to trade at this price because of the lackof counterparty from other participants. Our underlying idea for the relevance of this mechanismis that it can provide the best of both worlds, interpolating between CLOB and periodic auctions,by conveying information about a potential price change to all market participants in a timelymanner. On the one hand, auction phases enable market participants to source liquidity througha competitive process of price formation. On the other hand, potential local volume disequilibriabetween the needs of buyers and sellers that do not warrant a price change can be mitigated duringthe continuous sessions.Note that we focus here on AHEAD implementation on non-fragmented markets, such as small andmid-cap markets (some of these stocks may display a limited fragmentation, however discussionstowards the revision of MiFID II in Europe indicate the will of the regulator to impose a uniquestructure for such assets). In this case, AHEAD could be a decisive model since it improves liquid-ity aggregation. Furthermore, an auction on an illiquid instrument ending without any transactionwould still deliver a change in the clearing price of the instrument. Direct listings, where buildingsteadily liquidity is the key success factor, represent another situation where AHEAD could proveworthwhile and allow easier access to the financial markets for the small and mid-cap enterprises.Competition between two AHEAD markets could result in something closer to a more stable versionof a CLOB: in phases where both venues can trade on a fixed price, there would be situations whereone venue would display liquidity at a “bid” and the other venue at an “offer” (depending on thechosen make-take fees schedule) . In highly fragmented markets, CLOBs and auctions interact bycatering to different strategies from market participants. It would be very complex to model suchinteractions if AHEAD were to be added to the current microstructure, since the market participantmix would probably differ considerably from one venue to another. Such study is left for furtherresearch.We consider three agents in our model: two investors, one buyer and one seller, using aggressiveorders and one market maker using passive orders. The market maker provides liquidity during It would then be critical for the regulator to prevent a “race to the bottom” between competing venues by settingminimum values for the triggering quantities and the auction durations, as MiFID II did for the tick size. . Our buyer (resp. seller) investor wishes tobuy (resp. sell) a given amount of shares over a given time period. More specifically, we considerthat he aims at following a trading intensity target (coming for example from an Almgren-Chrisstype algorithm, see Almgren and Chriss (2001)). Thus his goal is to optimise his PnL while stayingclose to the target. From a mathematical viewpoint, his objective function consists into two termsthat he wants to minimise: one measuring his realized trading costs and the other the deviationfrom the target. To achieve their goal, our investors have access to two controls: the trading ratewith which they send their market orders and the triggering times of the auctions. They optimisesimultaneously and without communication their strategies. Note that there are of course morethan two investors in an actual market. However, since our auction period will be quite short, weexpect in practice only a small number of investors to take part in each auction (these investorsbeing probably different from one auction to the other). Note also that, in a live market environ-ment, participants are not restricted to aggressive orders and also compete through passive orders:in an AHEAD market, an aggressive order greater than the liquidity waiting for execution in theorder book would become a passive order for the remainder of the order.In this model, we show that the market admits a Nash equilibrium. This implies that ad-hoc auctions are a viable design as a trading mechanism. Furthermore, from our theoretical results,we can build a numerical methodology enabling us to compute the optimal strategies and valuefunctions of the investors under various market configurations. This is not only done in the ad-hoc auction framework but also under CLOB and periodic auction markets. This allows us to providea quantitative assessment of the AHEAD market from the investors’ viewpoint and to compare itwith the CLOB and periodic auction structures. Our main findings are the following. First AHEAD seems to be systematically preferable thanCLOBs from a market taker perspective. This is somehow in line with the results in Budish et al.(2015); Jusselin et al. (2019) which underline the relevance of auctions compared to CLOBs. Fur-thermore, based on our computations of the value functions, we conclude that for a large investor, ad-hoc auctions are always a suitable design (even compared with periodic auctions), in particularwhen the other investor is smaller. It enables the large investor to execute part of his orders with In a further study, the model could be developed to allow the market maker to manage its inventory by triggeringauctions himself and investors to use both passive and aggressive orders. ad-hoc andperiodic auctions. Essentially, if a small investor is still large enough to be able to trigger auctionswithout too much relative cost, the ad-hoc auction mechanism is beneficial for him. Otherwise,periodic auctions are more attractive from this investor’s viewpoint. In practice, in an actual mar-ket, the smaller investor could in fact even place passive orders and hence profit from the marketimpact generated by the larger one. Therefore a very small investor may prefer periodic auctionson instruments with high price viscosity/long queuing time because, in that case, the larger onecannot benefit from the continuous phase to reduce his volume imbalance in comparison to thesmaller investor, leading to very favourable auction clearing prices for the latter.The paper is organised as follows. In Section 2 we describe the ad-hoc auction mechanism and ourmodel. We introduce in Section 3 the notion of equilibrium in our framework and provide resultsabout the existence of such equilibrium under various types of assumptions. Numerical experimentsand economic insights can be found in Section 4. The proofs are relegated to the Appendix.
In this section, we introduce our model for a market with ad-hoc auctions. We build our mathe-matical framework and explain how our market participants (the two market takers and the marketmaker) interact. Then we describe the objectives of those participants in terms of optimisationproblems.
Let
T > h > c the set of continuous functionsfrom [0 , T + h ] into R , Ω d the set of piece-wise constant c`adl`ag functions from [0 , T + h ] into N ,and Ω = Ω c × (Ω d ) with corresponding Borel algebra F . The observable state is the canonicalprocess ( W t , ˜ N at , ˜ N bt ) t ∈ [0 ,T + h ] on the measurable space (Ω , F ) defined for any t ∈ [0 , T + h ] and ω =( w, n a , n b ) ∈ Ω by W t ( ω ) := w ( t ) , ˜ N at ( ω ) := n a ( t ) , ˜ N bt ( ω ) := n b ( t ) , with canonical completed filtration F = ( F t ) t ∈ [0 ,T + h ] = ( F ct ⊗ ( F dt ) ⊗ ) t ∈ [0 ,T + h ] .5he trading universe is reduced to a single risky asset with observable efficient price P ∗ given by P ∗ t := P ∗ + σW t , t ∈ [0 , T + h ] , with initial price P ∗ > σ >
0. The probability measure on Ω will bedefined so that W is a Brownian motion. The efficient price is to be understood as a benchmarkprice that market participants use to measure their trading costs by comparing it with the pricethey get in their actual transactions, see for example Delattre et al. (2013); Robert and Rosenbaum(2011); Stoikov (2018). The processes ˜ N a and ˜ N b will correspond to the quantities of orders sentby our two investors. We consider two investors (market takers) sending aggressive orders only. We call them Player a and Player b . Player a only sends buy market orders while Player b only sends sell market orders.Let λ − > λ + > λ − the maximum intensity. Weequip our filtered space with the probability P W ⊗ P N where P W is the Wiener measure and P N isthe solution to the martingale problem (in the sense of Jacod and Shiryaev (1987)) M t = ( ˜ N at , ˜ N bt ) T − t L with L = ( λ , λ ) T , 0 < λ < λ + , t ∈ [0 , T + h ]on ((Ω d ) , B ((Ω d ) ) , (( F dt ) ⊗ ) t ∈ [0 ,T + h ] ).In our model, Player a and Player b control the intensities of buy and sell orders respectively. Theset of admissible controls denoted by U is defined by all predictable processes with values in [ λ − , λ + ].For any pair ( λ a , λ b ) of admissible controls, we associate P λ a ,λ b the measure defined by d P λ a ,λ b d P (cid:12)(cid:12)(cid:12)(cid:12) t = Ψ λ a ,λ b t , where Ψ λ a ,λ b t is the Doleans-Dade exponential martingale given byΨ λ a ,λ b t = exp (cid:16) (cid:90) t (cid:0) log( λ as λ ) d ˜ N as − ( λ as − λ ) ds + log( λ bs λ ) d ˜ N bs − ( λ bs − λ ) ds (cid:1)(cid:17) . Thus, under the measure P λ a ,λ b , the processes ( ˜ N as − (cid:82) s λ au du ) ≤ s ≤ T + h , ( ˜ N bs − (cid:82) s λ bu du ) ≤ s ≤ T + h aremartingales and ( W s ) ≤ s ≤ T is still a Brownian motion independent of the processes ( ˜ N a , ˜ N b ). Inthe following, we denote by E λ a ,λ b the expectation under P λ a ,λ b and we write Ψ λ a ,λ b s,t = Ψ λ a ,λ b t / Ψ λ a ,λ b s for s ≤ t . 6he market takers can trigger an auction and we focus on analysing the market and the behavioursof the participants until the end of the auction. We do not consider successive auction phases asit would lead to important additional technical difficulties. Furthermore, we may expect that inpractice, under AHEAD, the market would be quite regenerative from one phase to the other. Wewrite T s,t with 0 ≤ s ≤ t ≤ T + h for the set of stopping times taking values in [ s, t ] and denote by τ a and τ b in T ,T the stopping times chosen by Player a and Player b respectively. An auction startsat time τ = τ a ∧ τ b , considering that if no player triggers an auction before time T , an auction isautomatically triggered at time T .Let ( τ, ˜ τ ) ∈ T ,T + h be such that τ ≤ ˜ τ , P − a.s. and λ ∈ U . We denote by λ [ τ, ˜ τ ] and U [ τ, ˜ τ ] therestriction of λ , respectively U , to [ τ, ˜ τ ]. For any λ ∈ U and µ ∈ U [ τ,T + h ] , we set ( λ ⊗ τ µ ) u := λ u u ≤ τ + µ u τ
In practice, in a continuous-time market, the two players would of course nevertrigger an auction at the same time as the matching engine needs anyway to process one messagefirst. It is actually a straightforward extension to consider the case where for Player a , ˆ n ab isreplaced by a random variable taking values or ˆ n with probability . and for Player b by ˆ n minusthis variable. We will actually consider such situation in the numerical results of Section 4 but keep n ab for simplicity for the theoretical developments. In addition, note that we can very well think of asituation where the exchange would let participants trigger auctions only at some (frequent) specifictimes. Let P ∈ R be a price fixed at t = 0. During the continuous phase, at time t , the market makeraccepts an order from Player a (buy order) if P > P ∗ t . In this case, a unit quantity is traded at price P . Symmetrically, he accepts an order from Player b (sell order) if P < P ∗ t and then a unit quan-tity is traded at price P . In other words, at time t during the continuous trading phase, Player a pays P P >P ∗ t d ˜ N at to buy P >P ∗ t d ˜ N at , while Player b earns P P
P ∗ s + s>τ ) d ˜ N as , N bt = (cid:90) t ( s ≤ τ P
τ ) d ˜ N bs . During the auction, the market maker is willing to buy or sell a given quantity at a certain price.We consider that he provides a mid-price, that we naturally take equal to P ∗ τ + h and a slope K ∈ R ,meaning that he offers a volume K ( p − P ∗ τ + h ) at time τ + h when the auction price is p ∈ R . Player a sends N aτ + h − N aτ + N a + buy market orders during the auction and Player b sends N bτ + h − N bτ + N b + sell market orders. So and similarly to Jusselin et al. (2019), the auction clearing price P auc fixedat the clearing time τ + h is solution of the equation which equals supply and demand:0 = − K ( P auc − P ∗ τ + h ) + ( N aτ + h − N aτ + N a + ) − ( N bτ + h − N bτ + N b + ) i.e. P auc = P ∗ τ + h + ( N aτ + h − N aτ + N a + ) − ( N bτ + h − N bτ + N b + ) K . (1)Thus, at the end of the auction, Player a buys N aτ + h − N aτ + N a + units at price P auc and Player b sells N bτ + h − N bτ + N b + units at price P auc . Remark 2.2.
One could think the market maker should rather take a mid-price equal to ±∞ if N aτ + h − N aτ + N a + ≶ N bτ + h − N bτ + N b + to optimise his PnL. However, in a real market, market makerssend limit orders over a bounded price interval and competition between them prevents them from isplaying irrealistic prices. Also, there is in practice uncertainty on the traded volumes (notablybecause auctions durations are slightly randomised). High uncertainty would lead to a high value of K to compensate the lack of information on ( N aτ + h − N aτ + N a + ) − ( N bτ + h − N bτ + N b + ) . Both market takers wish to optimise their PnL per unit of time. We suppose that they comparethe prices they get to the efficient price P ∗ seen as a benchmark. Moreover, they aim at trading acertain number of assets per unit of time (respectively v a and v b units per second) and have to paypenalties if they do not reach those targets. We now give an explicit decomposition of their tradingcosts per unit of time. As explained above, we assume that our two players are penalised during the continuous marketphase if they do not trade the right volumes. More precisely, during the continuous market phase,we consider the costs of Player a and Player b are respectively given for any t ∈ [0 , T + h ] by L at = q (cid:90) t ∧ τ ( v a s − N as ) ds + (cid:90) t ∧ τ ( P − P ∗ t ) P >P ∗ t dN at L bt = q (cid:90) t ∧ τ ( v b s − N bs ) ds − (cid:90) t ∧ τ ( P − P ∗ t ) P
0, represents the penalty if the number of tradesdoes not match the targeted value and the second one is the cost resulting from trading activitiescompared to the benchmark price P ∗ . Remark 2.3.
We could also compute the actual trading costs instead of the costs with respect tothe efficient price, replacing P − P ∗ t by P in (2) . We now turn to the costs Player a and Player b are subjected to during the auction. We assumeagain that both players are penalised during the auction if they do not trade at the rate v a and v b respectively. The penalty here is also quadratic with parameter q >
0. Thus the penalties of Player a and Player b during the auction are respectively given by C aauc = qh ( v a ( τ + h ) − N aτ + h − N a + ) and C bauc = qh ( v b ( τ + h ) − N bτ + h − N b + ) .
9s in Jusselin et al. (2019), the cost resulting from trading activities of market taker a is givenby N aτ,τ + h ( P auc − P ∗ τ + h ) while the gain of b resulting from his trades is N bτ,τ + h ( P auc − P ∗ τ + h ), where N aτ,τ + h = N aτ + h − N aτ + N a + and N bτ,τ + h = N bτ + h − N bτ + N b + .Putting together all the costs/gains of our market takers and using Equation (1), we get that thetotal cost of Player a per unit of time is given by L aτ + C aauc + N aτ,τ + h ( P auc − P ∗ τ + h ) τ + h = L aτ + C aauc + N aτ,τ + h ∆ N τ,τ + h K τ + h while the gain of Player b is − L bτ − C bauc + N bτ,τ + h ( P auc − P ∗ τ + h ) τ + h = − L bτ − C bauc + N bτ,τ + h ∆ N τ,τ + h K τ + h , where we set ∆ N τ,τ + h = N aτ,τ + h − N bτ,τ + h . For x ∈ R + × N × N × R × R and a pair of controls(( τ a , λ a ) , ( τ b , λ b )), let J a ( x , ( τ a , λ a ) , ( τ b , λ b )) = E λ a ,λ b (cid:2) L aτ + C aauc + N aτ,τ + h ∆ N τ,τ + h K τ + h (cid:12)(cid:12) ( P ∗ , N a , N b , L a , L b ) = x (cid:3) (3)and J b ( x , ( τ a , λ a ) , ( τ b , λ b )) = E λ a ,λ b (cid:2) − L bτ − C bauc + N bτ,τ + h ∆ N τ,τ + h K τ + h (cid:12)(cid:12) ( P ∗ , N a , N b , L a , L b ) = x (cid:3) . (4)Since Player a aims at minimising his cost, his goal is to minimise over ( τ a , λ a ) the objective function J a ( x , ( τ a , λ a ) , ( τ b , λ b )) (5)where ( τ b , λ b ) are controlled by Player b . Symmetrically, since Player b aims at maximising his gain,his goal is to maximise over ( τ b , λ b ) the objective function J b ( x , ( τ a , λ a ) , ( τ b , λ b )) (6)where ( τ a , λ a ) are controlled by Player a . Remark 2.4.
Using Lemma C.1 from Jusselin et al. (2019) and the fact that τ ≤ T and h > , weobtain that (3) and (4) are well-defined and finite. Nash equilibrium for pure and mixed stopping games
In this section, we investigate the existence of an equilibrium in the optimisation problems of themarket takers in the sense of Nash equilibrium adapted to our framework. We start by defining thenotion of open-loop Nash equilibrium in the sense of Carmona and Delarue (2018). Then we showthat restraining the set of stopping times to those taking values in a finite set allows us to buildan equilibrium in the simple case ˜ n = ˜ n ab = 0 and in the general case by considering generalizedstopping times. First we define the notion of open-loop Nash equilibrium.
Definition 3.1 (Open-Loop Nash Equilibrum (OLNE)) . Given x ∈ R + × N × N × R × R , we saythat the pair of controls (( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) is an open-loop Nash equilibrum of the game (OLNEfor short) if J a ( x , ( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) ≤ J a ( x , ( τ a , λ a ) , ( τ b, ∗ , λ b, ∗ )) ∀ ( τ a , λ a ) ∈ T ,T × U J b ( x , ( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) ≥ J b ( x , ( τ a, ∗ , λ a, ∗ ) , ( τ b , λ b )) ∀ ( τ b , λ b ) ∈ T ,T × U . We now define a Nash equilibrium for the auction phase.
Definition 3.2 (Open-loop Nash equilibrium for the τ − sub-game) . Given x ∈ R + × N × N × R × R and τ ∈ T ,T , we say that the pair of controls ( µ a, ∗ , µ b, ∗ ) is an open-loop Nash equilibrium for the τ -sub-game if E µ a, ∗ ,µ b, ∗ τ (cid:104) C aauc + N aτ,τ + h ∆ N τ,τ + h K (cid:105) = inf µ a ∈U [ τ,T + h ] E µ a ,µ b, ∗ τ (cid:104) C aauc + N aτ,τ + h ∆ N τ,τ + h K (cid:105) E µ a, ∗ ,µ b, ∗ τ (cid:104) − C bauc + N bτ,τ + h ∆ N τ,τ + h K (cid:105) = sup µ b ∈U [ τ,T + h ] E µ a, ∗ ,µ b τ (cid:104) − C bauc + N bτ,τ + h ∆ N τ,τ + h K (cid:105) , where E τ [ · ] := E [ ·| ( P ∗ τ , N aτ , N bτ , L aτ , L bτ ) = x ] . If an open-loop Nash equilibrium exists for the τ -sub-game, we write ξ aτ = E µ a, ∗ ,µ b, ∗ τ (cid:104) C aauc + N aτ,τ + h ∆ N τ,τ + h K (cid:105) ξ bτ = E µ a, ∗ ,µ b, ∗ τ (cid:104) − C bauc + N bτ,τ + h ∆ N τ,τ + h K (cid:105) (7)for the payoff of the sub-game (where a given open-loop Nash equilibrium for the τ -sub-game ischosen). 11imilarly to the results of Hamad`ene and Mu (2014); Jusselin et al. (2019), we know that thereexists an open-loop Nash equilibrium for the τ − sub-game (7). Thanks to a dynamic programmingargument, we can show that we can start by finding optimal controls for the sub-game startingat τ and that an OLNE for the game provides an open-loop Nash equilibrium for the sub-gamecorresponding to the auction phase. This is stated in the following proposition. Proposition 3.1.
Let x ∈ R + × N × N × R × R . For any τ ∈ T ,T , there exists at least oneopen-loop Nash equilibrium ( µ a, ∗ , µ b, ∗ ) to the τ -sub-game. Moreover, we can find two determinis-tic functions with polynomial growth g a , g b such that ξ aτ = g a ( N aτ − v a τ, N bτ − v b τ, N a + , N b + ) and ξ bτ = g b ( N aτ − v a τ, N bτ − v b τ, N a + , N b + ) .Finally, if (( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) is an OLNE for the general game, then the following dynamicprogramming principle holds J a ( x , ( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) = inf τ a ∈T ,T ,λ a ∈U [0 ,τ ] E λ a ,λ b, ∗ (cid:2) L aτ + ξ aτ τ + h (cid:3) J b ( x , ( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) = sup τ b ∈T ,T ,λ b ∈U [0 ,τ ] E λ a, ∗ ,λ b (cid:2) − L bτ + ξ bτ τ + h (cid:3) (8) where τ = τ a, ∗ ∧ τ b, ∗ and ( λ a, ∗ [ τ,T + h ] , λ b, ∗ [ τ,T + h ] ) is an open-loop Nash equilibrium for the τ -sub-game (7) (with payoffs ξ aτ and ξ bτ ), recalling that ( λ a, ∗ [ τ,T + h ] , λ b, ∗ [ τ,T + h ] ) is the restriction of ( λ a, ∗ , λ b, ∗ ) to [ τ, T + h ] .Proof. See Appendix A.It will be useful to consider the functions defined on [0 , T ] × N × N and for l =a or l =b associatedto the value of the sub-game for Player l when • he initiates the auction alone: g first l ( s, n a , n b ) = g l ( n a − v a s, n b − v b s, ˆ n l = a , ˆ n l = b ) , • he does not initiate the auction: g second l ( s, n a , n b ) = g l ( n a − v a s, n b − v b s, ˆ n l = b , ˆ n l = a ) , • he initiates it at the same time as the other player: g sim l ( s, n a , n b ) = g l ( n a − v a s, n b − v b s, ˆ n ab , ˆ n ab ) , the auction starts at T : g T l ( s, n a , n b ) = g l ( n a − v a s, n b − v b s, , . Note that from Proposition 3.1, we know that these functions have polynomial growth.
Remark 3.1.
To build an OLNE for the general game, we can start by building an equilibrium forthe sub-game (7) during the auction and then look for a solution to the problem (8) . Remark 3.2.
The uniqueness of an open-loop Nash equilibrium for the τ − sub-game (7) playedduring the auction is known to be a very intricate issue, see Hamad`ene and Mu (2014); Jusselin et al.(2019). However, numerical experiments seem to indicate that there is only one Nash equilibriumfor the sub-game for each value of ( P ∗ τ , N aτ − v a τ, N bτ − v b τ, N a + , N b + ) . Extending the results of A¨ıd et al. (2020) and Basei et al. (2019) to include jump processes andexpectations given by non-trivial risk measures, we can prove that the existence of an OLNE canbe reduced to solving a system of fully coupled integro-partial PDEs. We refer to Appendix B formore details on it. However, we do not expect to obtain the existence of an OLNE in this case ina general setting. Nevertheless, as we will see below, assuming that the players can choose theirstopping time only in a set of discrete times allows us to derive the existence of a Nash equilibriumin this slightly simplified setting.From now on, we focus on stopping times with values in a discrete subset of [0 , T ]. We look for an OLNE in the case where the stopping times can only take discrete values. Set δ ∈ R + such that Tδ ∈ N . For k = 0 , ..., Tδ , we consider T dkδ,T the set of stopping times with valuesin the set { kδ, ( k + 1) δ, ..., T } almost surely. Note that T d ( k +1) δ,T is included in T dkδ,T . Remark 3.3.
The following results can be easily extended to the case where the stopping times takevalues in any finite discrete set.
For any l ∈ (cid:74) , T /δ − (cid:75) and ( λ k ) k ∈ (cid:74) l,T/δ − (cid:75) ∈ U [ lδ, ( l +1) δ ] × ... × U [ T − δ,T ] , we set (cid:79) k ∈ (cid:74) l,T/δ − (cid:75) λ k := lδ ≤ t ≤ ( l +1) δ λ l + (cid:88) k ∈ (cid:74) l +1 ,T/δ − (cid:75) kδ Let k ∈ (cid:74) , T /δ (cid:75) , ˆ g ak and ˆ g bk be two measurable functions defined on [0 , T + h ] × R × R × N to R , with polynomial growth. Then, their exists ( λ ak − , λ bk − ) ∈ U k − δ,kδ ] such that E λ ak − ,λ bk − ( k − δ (cid:2) ˆ g ak ( kδ, P ∗ kδ , L akδ , L bkδ , N akδ , N bkδ ) (cid:3) = ess inf λ a ∈U [( k − δ,kδ ] E λ a ,λ bk − ( k − δ (cid:2) ˆ g ak ( kδ, P ∗ kδ , L akδ , L bkδ , N akδ , N bkδ ) (cid:3) E λ ak − ,λ bk − ( k − δ (cid:2) ˆ g bk ( kδ, P ∗ kδ , L akδ , L bkδ , N akδ , N bkδ ) (cid:3) = ess sup λ b ∈U [( k − δ,kδ ] E λ ak − ,λ b ( k − δ (cid:2) ˆ g bk ( kδ, P ∗ kδ , L akδ , L bkδ , N akδ , N bkδ ) (cid:3) . In the spirit of Ludkovski (2010), we will show the following results: • In the case where the triggering cost is null or small enough so that it does not impact thestrategies during the auction, see Section 3.2.2, we can always construct a Nash equilibriumby backward induction (see Theorem 3.1). • In the general case, see Section 3.2.3, we can construct a Nash equilibrium if we extend ourprobability space to allow for randomised stopping times, see Theorem 3.2. • In both cases, the pure open-loop Nash equilibrium or randomised open-loop Nash equilibriumof the discretised game is an (cid:15) -Nash equilibrium of the continuous game, see Section 3.2.4. We now introduce the notion of discretised game and that of Nash equilibrium in this framework. Definition 3.3 (Pure Open-Loop Nash Equilibrium for the discrete game (OLNED)) . Let x ∈ R + × N × N × R × R . We say that (( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) ∈ ( T d ,T × U ) is a pure open-loop Nashequilibrium of the discretised game (OLNED for short) if it is a solution to the game E λ a, ∗ ,λ b, ∗ (cid:104) L aτ + C aauc + N aτ,τ + h ∆ N τ,τ + h K τ + h (cid:105) = inf τ a ∈T d ,T ,λ a ∈U E λ a ,λ b, ∗ (cid:104) L a ˜ τ a + C aauc + N a ˜ τ a , ˜ τ a + h ∆ N ˜ τa, ˜ τa + h K ˜ τ a + h (cid:105) E λ a, ∗ ,λ b, ∗ (cid:104) − L bτ − C bauc + N bτ,τ + h ∆ N τ,τ + h K τ + h (cid:105) = sup τ b ∈T d ,T ,λ b ∈U E λ a, ∗ ,λ b (cid:104) − L b ˜ τ b − C bauc + N b ˜ τ b , ˜ τ b + h ∆ N ˜ τb, ˜ τb + h K ˜ τ b + h (cid:105) a.s., with ˜ τ a = τ a ∧ τ b, ∗ , ˜ τ b = τ a, ∗ ∧ τ b , τ = τ a, ∗ ∧ τ b, ∗ and where E [ · ] := E [ ·| ( P ∗ , N a , N b , L a , L b ) = x ] . For sake of simplicity, a pure OLNED will be simply called an OLNED. Inspired by the literature onoptimal stopping in non-zero sum games in discrete time, see among others Grigorova and Quenez(2017); Riedel and Steg (2017), we aim at finding OLNED in the sense of the above definition.14 .2.2 Particular case: ˆ n = ˆ n ab = 0 , no cost to trigger the auction We first consider the simple case ˆ n = ˆ n ab = 0. In this situation, there is no cost associated with thetriggering of an auction and market takers are indifferent about stopping the game first or second.We will see that here a Nash equilibrium for the discretised game can be constructed explicitly.For ( τ a , τ b ) ∈ ( T d ,T ) , if Player b ’s stopping time is τ b , then the value of Player a is the same whetherhe plays τ a or τ a ∧ τ b . Symmetrically, the value of Player b is the same whether he plays τ b or τ a ∧ τ b . So we can simply consider strategies where both players stop at the same time. We lookfor a stopping time τ ∗ ∈ T d ,T and trading intensities ( λ a, ∗ , λ b, ∗ ) ∈ U such that J a ( x , ( τ ∗ , λ a, ∗ ) , ( τ ∗ , λ b, ∗ )) ≤ J a ( x , ( τ a , λ a ) , ( τ ∗ , λ b, ∗ )) ∀ ( τ a , λ a ) ∈ T d ,T × U J b ( x , ( τ ∗ , λ a, ∗ ) , ( τ ∗ , λ b, ∗ )) ≥ J b ( x , ( τ ∗ , λ a, ∗ ) , ( τ b , λ b )) ∀ ( τ b , λ b ) ∈ T d ,T . × U for x ∈ R + × N × N × R × R .We build by backward induction a process ( U ak , U bk ) k ∈ (cid:74) ,T/δ (cid:75) , adapted to the discrete filtration( F δk ) k ∈{ , ,...,T/δ ∈ N } so that, for each k ∈ { , , ..., T /δ ∈ N } , U ak and U bk are the values of Player a and Player b at time kδ when they both play an OLNED. Backward induction algorithm. We formally construct by backward induction an OLNED.We start by setting( U aT/δ , U bT/δ ) = ( L aT + g first a ( T, N aT , N bT ) T + h , − L bT + g first b ( T, N aT , N bT ) T + h )and τ ∗ Tδ = T. Since the players are forced to enter an auction if they have not started one before time T andbecause ˆ n = ˆ n ab = 0, these values are those of the game if the players start playing at time T . Inthis case, they play a Nash equilibrium during the auction denoted by (ˆ λ a , ˆ λ b ) by solving (7), whichwe can compute with the same numerical method as in Jusselin et al. (2019).In the interval ( T − δ, T ], the players cannot trigger an auction. From Lemma 3.1, we find15 λ a, ∗ Tδ − , λ b, ∗ Tδ − ) ∈ U T − δ,T ] such that E λ a, ∗ Tδ − ,λ b, ∗ Tδ − T − δ (cid:2) U aT/δ (cid:3) = ess inf λ a ∈U [ T − δ,T ] E λ a ,λ b, ∗ Tδ − T − δ (cid:2) U aT/δ (cid:3) E λ a, ∗ Tδ − ,λ b, ∗ Tδ − T − δ (cid:2) U bT/δ (cid:3) = ess sup λ b ∈U [ T − δ,T ] E λ a, ∗ Tδ − ,λ b T − δ (cid:2) U bT/δ (cid:3) . We set ˜ λ a, ∗ Tδ − := λ a, ∗ Tδ − ⊗ T ˆ λ a and ˜ λ b, ∗ Tδ − := λ b, ∗ Tδ − ⊗ T ˆ λ b .At time T − δ , both players can choose whether to trigger an auction or not. Also, they are indifferentabout who actually triggers the auction. If one of the players triggers an auction the values become( L aT − δ + g first a ( T − δ,N aT − δ ,N bT − δ ) T − δ + h , − L bT − δ + g first b ( T − δ,N aT − δ ,N bT − δ ) T − δ + h ). Otherwise, if none of the players triggers anauction, their values are ( E λ a, ∗ Tδ − ,λ b, ∗ Tδ − T − δ (cid:2) U aT/δ (cid:3) , E λ a, ∗ Tδ − ,λ b, ∗ Tδ − T − δ (cid:2) U bT/δ (cid:3) ). So each player compares the twopossible values ( i.e. the two possible mean payoffs) and triggers an auction if and only if it isbeneficial to him. Consequently, if the following condition is satisfied: E λ a, ∗ Tδ − ,λ b, ∗ Tδ − T − δ (cid:2) U aT/δ (cid:3) < L aT − δ + g first a ( T − δ,N aT − δ ,N bT − δ ) T − δ + h E λ a, ∗ Tδ − ,λ b, ∗ Tδ − T − δ (cid:2) U bT/δ (cid:3) > − L bT − δ + g first b ( T − δ,N aT − δ ,N bT − δ ) T − δ + h . , then none of the players triggers an auction and we set( U aT/δ − , U bT/δ − ) = ( E λ a, ∗ Tδ − ,λ b, ∗ Tδ − T − δ (cid:2) U aT/δ (cid:3) , E λ a, ∗ Tδ − ,λ b, ∗ Tδ − T − δ (cid:2) U bT/δ (cid:3) )and τ ∗ Tδ − = τ ∗ Tδ . Otherwise( U aT/δ − , U bT/δ − ) = ( L aT − δ + g first a ( T − δ, N aT − δ , N bT − δ ) T − δ + h , − L bT − δ + g first b ( T − δ, N aT − δ , N bT − δ ) T − δ + h ) , in which case the players trigger an auction at T − δ and so τ ∗ Tδ − = T − δ. Then we iterate the procedure to build U a and U b and ( τ ∗ , ˜ λ a, ∗ ) , ( τ ∗ , ˜ λ b, ∗ ) at any discrete time:Using again backward induction, we can show that, for every k ∈ (cid:74) , T /δ (cid:75) , there exist two functions g ak and g bk such that U ak = ˆ g ak ( kδ, P ∗ kδ , L akδ , L bkδ , N akδ , N bkδ ) and U bk = ˆ g bk ( kδ, P ∗ kδ , L akδ , L bkδ , N akδ , N bkδ ). It16s indeed true for k = T /δ . Then, using a result from Jusselin et al. (2019), we have that if thisproperty holds for some k ∈ (cid:74) , T /δ (cid:75) , it also holds for k − 1. This allows us to apply the previousmethodology and find appropriate ( λ a, ∗ k , λ b, ∗ k ) on each interval. As a result ( τ ∗ , ˜ λ a, ∗ ) , ( τ ∗ , ˜ λ b, ∗ ) is anOLNED. This backward induction is summed up in Algorithm 1, see Appendix D. Existence of an OLNED. The following theorem formalizes this procedure. Theorem 3.1. Let U a , U b , λ a, ∗ , λ b, ∗ be defined by the backward induction in Algorithm 1 and set τ ∗ = δ inf (cid:110) l ∈ (cid:74) , T /δ (cid:75) , U al = L alδ + g first a ( lδ, N alδ , N blδ ) lδ + h or U bl = − L blδ + g first b ( lδ, N alδ , N blδ ) lδ + h (cid:111) . Let ˜ λ a, ∗ = (cid:0) (cid:78) l ∈ (cid:74) ,T/δ − (cid:75) λ a, ∗ l (cid:1) ⊗ τ ∗ ˆ λ a and ˜ λ b, ∗ = (cid:0) (cid:78) l ∈ (cid:74) ,T/δ − (cid:75) λ b, ∗ l (cid:1) ⊗ τ ∗ ˆ λ b . Then, the pair of controls (( τ ∗ , ˜ λ a, ∗ ) , ( τ ∗ , ˜ λ b, ∗ )) is an OLNED. In this case, U a and U b are the values of the discretised gamefor Player a and Player b respectively, i.e. U a = inf τ a ∈T d ,T ,λ a ∈U E λ a , ˜ λ b, ∗ (cid:2) L aτ + C aauc + N aτ,τ + h ( N aτ,τ + h − N bτ,τ + h ) K τ + h (cid:3) where τ = τ a ∧ τ ∗ and U b = sup τ b ∈T d ,T ,λ b ∈U E ˜ λ a, ∗ ,λ b (cid:2) − L bτ − C bauc + N bτ,τ + h ( N aτ,τ + h − N bτ,τ + h ) K τ + h (cid:3) where τ = τ ∗ ∧ τ b .Proof. See Appendix C.1 Remark 3.4. Note that the condition ˆ n = ˆ n ab = 0 is sufficient but not necessary to obtain Theorem3.1. A weaker condition is actually g first i = g second i = g sim i for i = a, b . Note that at each time kδ , k ∈ (cid:74) , Tδ − (cid:75) , the players deal with a 2 × × n = ˆ n ab = 0)is much more intricate to get. However, we can obtain such result if we consider randomisedstrategies. This leads us to the notion of randomised discrete stopping times as explained below. We now consider the general case. The procedure used to build a Nash equilibrium in Section 3.2.2can be adapted to construct a Nash equilibrium in the general case. This can be done if we look17 b stops continuesstops ( L akδ + g first a kδ + h , − L bkδ + g first b kδ + h ) ( L akδ + g first a kδ + h , − L bkδ + g first b kδ + h )continues ( L akδ + g first a kδ + h , − L bkδ + g first b kδ + h ) ( E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) , E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3) )Table 1: Cost/gain for Player a / b depending on whether Player a / b stops or not the discrete gameplayed at time kδ .for generalized stopping times instead of classical stopping times. We refer to Coquet and Toldo(2007); Solan et al. (2012); Touzi and Vieille (2002) for various optimal stopping problems dealingwith this kind of stopping times. Informal derivation of a mixed Nash equilibrium. Right after T − δ and until T , the situationis the same as in the case where ˆ n = ˆ n ab = 0. The players cannot trigger an auction so they playa Nash equilibrium (with no stopping allowed) until T . In that case, their values right after T − δ are ( E λ a, ∗ ,λ b, ∗ T − δ (cid:2) U aT/δ (cid:3) , E λ a, ∗ ,λ b, ∗ T − δ (cid:2) U bT/δ (cid:3) ). At time T − δ , they are allowed to trigger an auction. Thepayoffs depend now on which player triggers an auction. • If Player a triggers an auction and Player b does not, the values are( L aT − δ + g first a ( T − δ, N aT − δ , N bT − δ ) T − δ + h , − L bT − δ + g second b ( T − δ, N aT − δ , N bT − δ ) T − δ + h ) . • If Player b triggers an auction and Player a does not, the values are( L aT − δ + g second a ( T − δ, N aT − δ , N bT − δ ) T − δ + h , − L bT − δ + g first b ( T − δ, N aT − δ , N bT − δ ) T − δ + h ) . • If both players trigger an auction, the values are( L aT − δ + g sim a ( T − δ, N aT − δ , N bT − δ ) T − δ + h , − L bT − δ + g sim b ( T − δ, N aT − δ , N bT − δ ) T − δ + h ) . • Finally, if none of the players trigger an auction the values are( E λ a, ∗ ,λ b, ∗ T − δ (cid:2) U aT/δ (cid:3) , E λ a, ∗ ,λ b, ∗ T − δ (cid:2) U bT/δ (cid:3) ) . Contrary to the previous case, there is some advantage to gain when the other player triggers anauction. Let p i Tδ − = 1 if Player i = a, b triggers an auction at T − δ and 0 otherwise. For p b Tδ − p a Tδ − must be a minimiser of p ∈ { , } (cid:55)−→ p p b Tδ − L aT − δ + g sim a T − δ + h + p (1 − p b Tδ − ) L aT − δ + g first a T − δ + h + (1 − p ) p b Tδ − L aT − δ + g second a T − δ + h + (1 − p )(1 − p b Tδ − ) E λ a, ∗ k ,λ b, ∗ Tδ − T − δ (cid:2) U aT/δ (cid:3) , while for p a Tδ − fixed, p b Tδ − must be a maximiser of p ∈ { , } (cid:55)−→ p a Tδ − p − L bT − δ + g sim b T − δ + h + p a Tδ − (1 − p ) − L bT − δ + g second b T − δ + h + (1 − p a Tδ − ) p − L bT − δ + g first b T − δ + h + (1 − p a Tδ − )(1 − p ) E λ a, ∗ k ,λ b, ∗ Tδ − T − δ (cid:2) U bT/δ (cid:3) . Such optimisers might not always exist or might not be unique (in the sense that we would have todecide who triggers the auction). However, both players can always find probabilities of stopping p a and p b in [0 , 1] such that, if Player b triggers an auction with probability p b , the optimal probabilityof stopping for Player a is p a , and conversely, if Player a triggers an auction with probability p a ,the optimal probability of stopping for Player b is p b . Additionally, it is often more natural toconsider probabilities of stopping, in particular in the frequent case where both ( p a , p b ) = (1 , 0) and( p a , p b ) = (0 , 1) are possible pure Nash equilibria. This describes a mixed Nash equilibrium andsimply corresponds to a solution of the convexification of the above problem.There are multiple equivalent notions of random times which stop according to some probability.We use here the notion of mixed stopping times of Laraki and Solan (2005, 2010) to build ourprobability space. Definition 3.4 (Generalized stopping time) . A generalized stopping time is a measurable function µ : Ω × [0 , → [0 , T ] such that for Λ -almost every r ∈ [0 , , where Λ denotes the Lebesgue measure,the function ω → µ ( ω, r ) is a stopping time, i.e. µ ( ., r ) ∈ T ,T . Our probability space then becomes (Ω × [0 , × [0 , , P ⊗ Λ ⊗ Λ) where the first extension char-acterizes the randomiser of Player a ’s stopping time and the second one that of Player b ’s stoppingtime. Let 0 ≤ s ≤ t ≤ T . We denote by T ∗ s,t the set of generalized stopping times with values in[ s, t ]. If s/δ ∈ N and t/δ ∈ N , we also denote by T ∗ ,ds,t the set of generalized stopping times withvalues in (cid:74) s, t (cid:75) .We also extend the definition of Nash equilibrium in this framework. Definition 3.5 (Mixed OLNE and mixed OLNED) . Let x ∈ R + × N × N × R × R . We say that τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) ∈ ( T ∗ ,T × U ) × ( T ∗ ,T × U ) , resp. ( T ∗ ,d ,T × U ) × ( T ∗ ,d ,T × U ) , is a mixed OLNE,resp. mixed OLNED, if it is a solution to the game E λ a, ∗ ,λ b, ∗ (cid:104) L a ˜ τ a + C aauc + N a ˜ τa, ˜ τa + h ∆ N ˜ τa, ˜ τa + h K ˜ τ a + h (cid:105) = inf τ a ∈T ∗ ,T ,λ a ∈U E λ a ,λ b, ∗ (cid:104) L a ˜ τ a + C aauc + N a ˜ τa, ˜ τa + h ∆ N ˜ τa, ˜ τa + h K ˜ τ a + h (cid:105) E λ a, ∗ ,λ b, ∗ (cid:104) − L b ˜ τ b − C bauc + N b ˜ τb, ˜ τb + h ∆ N ˜ τb, ˜ τb + h K ˜ τ b + h (cid:105) = sup τ b ∈T ∗ ,T ,λ b ∈U E λ a, ∗ ,λ b (cid:104) − L b ˜ τ b − C bauc + N b ˜ τb, ˜ τb + h ∆ N ˜ τb, ˜ τb + h K ˜ τ b + h (cid:105) resp. E λ a, ∗ ,λ b, ∗ (cid:104) L a ˜ τ a + C aauc + N a ˜ τa, ˜ τa + h ∆ N ˜ τa, ˜ τa + h K ˜ τ a + h (cid:105) = inf τ a ∈T ∗ ,d ,T ,λ a ∈U E λ a ,λ b, ∗ (cid:104) L a ˜ τ a + C aauc + N a ˜ τa, ˜ τa + h ∆ N ˜ τa, ˜ τa + h K ˜ τ a + h (cid:105) E λ a, ∗ ,λ b, ∗ (cid:104) − L b ˜ τ b − C bauc + N b ˜ τb, ˜ τb + h ∆ N ˜ τb, ˜ τb + h K ˜ τ b + h (cid:105) = sup τ b ∈T ∗ ,d ,T ,λ b ∈U E λ a, ∗ ,λ b (cid:104) − L b ˜ τ b − C bauc + N b ˜ τb, ˜ τb + h ∆ N ˜ τb, ˜ τb + h K ˜ τ b + h (cid:105) with ˜ τ a = τ a ∧ τ b, ∗ , ˜ τ b = τ a, ∗ ∧ τ b , τ = τ a, ∗ ∧ τ b, ∗ and where E [ · ] := E P ⊗ Λ ⊗ Λ [ ·| ( P ∗ , N a , N b , L a , L b ) = x ] . It is known (see for example Shmaya and Solan (2014); Solan et al. (2012); Touzi and Vieille(2002)) that our notion of generalized stopping time is equivalent to the notion described in theinformal derivation of a mixed Nash equilibrium above, where, at time t , each player stops withsome probability based on the information F t . In particular, we can build a mixed OLNED usingthe same algorithm as in Section 3.2.2, see Algorithm 2 in Appendix D. Existence of a (mixed) OLNED. More formally, the following theorem based on the backwardinduction above provides the existence of a mixed OLNED. Theorem 3.2. Let τ a, ∗ ( ., r ) = δ inf (cid:8) l ∈ (cid:74) , T /δ (cid:75) , − (cid:81) lk =0 (1 − p al ) ≥ r ) (cid:9) τ b, ∗ ( ., r ) = δ inf (cid:8) l ∈ (cid:74) , T /δ (cid:75) , − (cid:81) lk =0 (1 − p bl ) ≥ r ) (cid:9) where p a and p b are the discrete F t -adapted processes given by Algorithm 2 in Appendix D. Let ˜ λ a, ∗ = (cid:78) ( λ a, ∗ l ) l ∈ (cid:74) ,T/δ − (cid:75) ⊗ τ ∗ ˆ λ a and ˜ λ b, ∗ = (cid:78) ( λ b, ∗ l ) l ∈ (cid:74) ,T/δ − (cid:75) ⊗ τ ∗ ˆ λ b , where the quantities on the r.h.s of theequalities are also given in Algorithm 2 in Appendix D. Then the strategies (( τ a, ∗ , ˜ λ a, ∗ ) , ( τ b, ∗ k , ˜ λ b, ∗ )) describe a mixed OLNED.Proof. According to Shmaya and Solan (2014) and Solan et al. (2012), optimising over the set ofgeneralized stopping times is equivalent to optimising over the set of adapted processes p a and p b kδ or continue playing until ( k + 1) δ .a b stops continuesstops ( L akδ + g sim a kδ + h , − L bkδ + g sim b kδ + h ) ( L akδ + g first a kδ + h , − L bkδ + g second b kδ + h )continues ( L akδ + g second a kδ + h , − L bkδ + g first b kδ + h ) ( E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) , E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3) )Table 2: Cost/gain for Player a / b depending on whether Player a / b stops or not the discrete gameplayed at time kδ .Thus, at time kδ , p ak and p bk are defined as solutions of the following linear optimisation problems.For p bk fixed, Player a chooses p ak ∈ arg inf p ∈ [0 , (cid:110) p p bk L akδ + g sim a kδ + h + p (1 − p bk ) L akδ + g first a kδ + h + (1 − p ) p bk L akδ + g second a kδ + h + (1 − p )(1 − p bk ) E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3)(cid:111) , while for p ak fixed, Player b chooses p bk ∈ arg sup p ∈ [0 , (cid:110) p ak p − L bkδ + g sim b kδ + h + p ak (1 − p ) − L bkδ + g second b kδ + h + (1 − p ak ) p − L bkδ + g first b kδ + h + (1 − p ak )(1 − p ) E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3)(cid:9) . From classical results, see for instance Von Neumann and Morgenstern (1947); Nash (1950), weknow that the two problems above can be solved simultaneously . Solving for a mixed equilibriumyields the following result: It would no longer be the case in general with p ∈ { , } , i.e. with pure stopping times, although it works ifˆ n = ˆ n ab = 0 as we have seen before. p ak = L akδ + g first a kδ + h − E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) − L akδ + g sim a kδ + h + L akδ + g first a kδ + h + L akδ + g second a kδ + h − E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) p bk = − L bkδ + g first b kδ + h − E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3) − − L bkδ + g sim b kδ + h + − L bkδ + g first b kδ + h + − L bkδ + g second b kδ + h − E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3) (9)and one can easily verify that the denominators are non-zero and that these values are in [0 , 1] whenthere is no pure Nash equilibrium. Remark 3.5. The notion of probability of stopping is quite convenient for numerical computationsas we can compute the value functions and the strategies by dynamic programming. ε -OLNE In this part, we explain that the previously introduced OLNEDs (see Definition 3.3) provide goodapproximations for OLNEs (see Definition 3.1), in the sense of ε -Nash equilibria.For technical reasons we need to slightly modify the definitions of C aauc and C bauc replacing v a τ and v b τ by (cid:100) v a τ − (cid:101) and (cid:100) v b τ − (cid:101) . All the previous results could be proved in this slightly modifiedsetting. This assumption is crucial in this section as it enables us to have that when v a δ ∈ N and v b δ ∈ N , the functions g i, first , g i, sim , for i = a, b take the same values if we replace t by (cid:100) tδ (cid:101) δ . Thiswill be a key element in the proof of the next theorems. Theorem 3.3 (OLNED and ε − OLNE) . Let δ > and (( τ a , λ a ) , ( τ b , λ b )) ∈ ( T d ,T × U ) × ( T d ,T × U ) be the strategies associated to a pure OLNED starting at with time-step δ . Let ε > . Then, for δ small enough such that v a δ ∈ N and v b δ ∈ N , E λ a ,λ b (cid:2) L aτ a ∧ τ b + ξ aτ a ∧ τ b τ a ∧ τ b + h (cid:3) ≤ inf τ ∈T ,T ,λ ∈U E λ,λ b (cid:2) L aτ ∧ τ b + ξ aτ ∧ τ b τ ∧ τ b + h (cid:3) + ε + ε a E λ a ,λ b (cid:2) − L bτ a ∧ τ b + ξ bτ a ∧ τ b τ a ∧ τ b + h (cid:3) ≥ sup τ ∈T ,T ,λ ∈U E λ a ,λ (cid:2) − L bτ a ∧ τ + ξ bτ a ∧ τ τ a ∧ τ + h (cid:3) − ε − ε b where ε a = 1 h sup λ a ∈U ,λ b ∈U E λ a ,λ b (cid:2) sup [0 ,T ] max (cid:0) max( g sim a ( t, N at , N bt ) , g second a ( t, N at , N bt )) − g firsta ( t, N at , N bt ) , (cid:1)(cid:3) nd ε b = 1 h sup λ a ∈U ,λ b ∈U E λ a ,λ b (cid:2) sup [0 ,T ] max (cid:0) g firstb ( t, N at , N bt ) − min( g sim b ( t, N at , N bt ) , g second b ( t, N at , N bt )) , (cid:1)(cid:3) . Proof. See Appendix C.2.Also, this theorem extends easily to the case of mixed OLNEs and mixed OLNEDs. Theorem 3.4 (Mixed OLNED and ε − mixed OLNE) . Let δ > and (( τ ad , λ a ) , ( τ bd , λ b )) ∈ ( T ∗ ,d ,T ×U ) × ( T ∗ ,d ,T × U ) be the strategies of a mixed OLNED starting at . Let ε > . Then, for δ smallenough such that v a δ ∈ N and v b δ ∈ N , E λ a ,λ b (cid:2) L aτ a ∧ τ b + ξ aτ a ∧ τ b τ a ∧ τ b + h (cid:3) ≤ inf τ ∈T ∗ ,T ,λ ∈U E λ,λ b (cid:2) L aτ ∧ τ b + ξ aτ ∧ τ b τ ∧ τ b + h (cid:3) + ε + ε a E λ a ,λ b (cid:2) − L bτ a ∧ τ b + ξ bτ a ∧ τ b τ a ∧ τ b + h (cid:3) ≥ sup τ ∈T ∗ ,T ,λ ∈U E λ a ,λ (cid:2) − L bτ a ∧ τ + ξ bτ a ∧ τ τ a ∧ τ + h (cid:3) − ε − ε b . (10) Proof. The proof is the same as the proof of Theorem 3.3.In practice and in our numerical experiments, the constants ε a and ε b are negligible and very oftenzero. This is because they are non-zero only when there is an advantage in triggering the auctionright before the other player. This is typically not the case, unless the player triggering the auctionbenefits from ˆ n to execute a large target volume that he could not fully do within the auctionbecause of its bounded intensity. Remark 3.6. The condition on δ is only technical and ensures that the changes in the targetshappen on the same grid as the optimisation (with mesh δ ). This condition would actually not berequired if the targets followed, for example, Poisson processes. ad-hoc auctions In this section, we provide numerical results enabling us to draw conclusions on the relevance of ad-hoc auctions compared to CLOB and periodic auctions. We also discuss some implementationdetails. The value functions shown are multiplied by 10 for more readability. First we show how the value of the sub-game played during the auction phase varies with theparameters, for Player a and Player b . From now on we take K = 10 , v a = v b = 0 . .1.1 Effect of the amount traded before the auction We fix h = 30 s and plot the value of the sub-game ξ aτ as a function of N aτ − v a τ , for various valuesof N a + , with N b + = 0 and N bτ − v b τ = 0 (see Figure 1, left side). First we notice that these graphsare increasing with respect to N a + , which is obviously not surprising. The effect of N a + gets moreimportant as N aτ − v a τ becomes larger. This is because in such situation, Player a is already inadvance regarding to his target. A large N a + implies that he is even more in advance and getspenalised via the objective function.Looking at the graphs for fixed N a + , we see that the best context to trigger an auction is when N at − v a t is close to zero and actually slightly negative. In that case, Player a can launch an auctionwithout overshooting his target at the end because of the mandatory volume N a + . Moreover, wenote that ξ aτ is large when either N aτ − v a τ is large, since Player a is penalised for overshooting histarget, or when N aτ − v a τ is too small, since Player a has to send a lot of orders during the auction,which makes the price increase.Next we plot ξ aτ as a function of N bτ − v b τ , for various values of N a + , with N b + = 0 and N aτ − v a τ = 0(see Figure 1, right side). We notice that ξ aτ is always increasing with respect to N bτ − v b τ : themore Player b trades before the auction in comparison to his target, the less he trades during theauction, and the higher the final price of the auction is. In addition to that, ξ aτ converges when N bτ − v b τ → ±∞ : if N bτ − v b τ is too large, Player b stops trading completely and if N bτ − v b τ istoo small, Player b would rather pay some penalties than send too many orders during the auctionleading to a bad price since Player a is at equilibrium when the auction starts.Figure 1: On the left, ξ aτ as a function of N aτ − v a τ , for different values of N a + , with N b + = 0, N bτ − v b τ = 0 and h = 30 s . On the right, ξ aτ as a function of N bτ − v b τ , for different values of N a + ,with N b + = 0, N aτ − v a τ = 0 and h = 30 s , q = 0 . .1.2 Impact of the risk aversion parameter and of the auction duration We investigate the effect of the parameter q which is the factor for the penalties received by theplayers for not reaching their trading targets and of the auction duration. We first consider q = 0 . ξ aτ as a function of h , for N aτ − v a τ = N bτ − v b τ = 0, N b + = 0and for multiple values of N a + . On the right side of Figure 2a, we display ξ aτ as a function of h , with N aτ − v a τ = N bτ − v b τ = 0, N a + = 0 and for multiple values of N b + . In Figure 2b we fix q = 0 . 01 andin Figure 2c q = 0 . ξ aτ = 0 for h large enough, which is no longer the case in Figure2c. This is because when the commitment to the target is severe, over a quite long time periodboth traders send on average the same number of orders as v a = v b and the effect of N a + or N b + vanishes. We also observe that too short auctions may create some kind of arbitrage opportunities:the trader who triggers an auction is committed to trade at least a given volume. The other tradermight choose to trade less to take advantage of the price imbalance in the auction, as the penalty hewill have to pay will not be too large. Let us take the example of h = 20 s . In that case, the targetis two lots for both Player a and Player b . If Player a triggers the auction with N a + = 4 then Player b will put a volume of 2 in the auction meeting his target or perhaps even less (volume of 1) meetingpartially his target but benefiting from price impact. Such phenomenon is magnified in a situationas in Figure 2c where the target commitment is very weak. In that case, both players try to benefitfrom price impact leading to a game where they both put smaller volumes than their target. For ex-ample, we see that the effect of the initial volume N a + = 1 vs N a + = 2 takes more than 80 seconds tovanish in Figure 2c, although the target is 8 lots for 80 s . This means that between 0 and 80 seconds,both investors play strategically to benefit from the effect of volume imbalance on the clearing price.This shows that the duration of the auction should be large enough and related to reasonablepractical values for q . Considering the auction duration helps to convey information to marketparticipants, it should also probably depend on the deviation between the previous clearing priceand the best offer price in the order book at the beginning of the auction. The larger the devia-tion, the longer the duration of the auction. Accurate duration calibration is left for further research.We use the results of this section to choose suitable parameters for our study of the entire ad-hoc auction in the next section. 25 a) q = 0 . q = 0 . q = 0 . Figure 2: ξ aτ as a function of h , with N aτ − v a τ = 0, N bτ − v b τ = 0. On the left, we fix N b + = 0 andconsider multiple values of N a + . On the right, we fix N a + = 0 and consider multiple values of N b + .26 .2 Assessment of ad-hoc auctions We now investigate the whole mechanism of ad-hoc auctions and compare it with the classicalCLOB and periodic auctions. We use Algorithm 2 with a small timestep δ = 0 . s and write V i for J i ( ., ( τ i, ∗ , λ i, ∗ ) , ( τ i, ∗ , λ i, ∗ )) for i = a, b . The values for v a and v b will be of order 0 . 1, so we expect roughly 2 trades every 10 seconds, whichcorresponds to the case of reasonably liquid assets. We fix T = 100 s so that T is large comparedto the average time between trades. We take q = 0 . , h = 20 s and ˆ n = n ab = 3. The justificationfor the relevance of these parameters is the following: • We have ˆ n > ( v a ∧ v b ) h . This ensures that transactions occur both in the continuous andauction phases. As a matter of fact, if ˆ n < ( v a ∧ v b ) h , the triggering cost for an auction isquite negligible with respect to the target amount within the auction. We numerically observethat in that case, investors do not use the continuous phase and trade only in the auctionphases, which means that ad-hoc auctions are reduced to periodic auctions. • Consider an auction triggered because both players are slightly behind their targets so thatone of them, say Player a , triggers the auction and both should trade 3 lots during the 20seconds. Then, suppose that Player b tries to benefit from the price impact and trade only 2during the auction instead of 3. Under these parameters, the price impact benefit of Player b (which is equal to 2 × /K ) is exactly the cost paid for not meeting the target (which isequal to qh ). Hence from the investors’ viewpoint, these parameters correspond to reasonablebalance between trading costs and target deviation penalties. ˆ n compared to v a and v b Here we replace ˆ n ab by a random variable which is so that if there is simultaneous triggering, itis attributed to Player a or Player b with probability 1 / of the continuous phase at time t = 0 for different values of v a and v b . Figure 3 : supply and demand of similar order. When v a = v b = 0 . 1, the average durationof the continuous phase is 21 seconds if the initial trading price P is equal to P ∗ . We observe thatwe obviously get a symmetric average duration of the continuous trading phase with respect to the In fact we only compute a proxy of the average duration. The computation details are given in Algorithm 3 inAppendix D.3. P − P ∗ . The average duration of the continuous phase is maximal at P = P ∗ , then decreasesand becomes stationary. This can be explained as follows: if P is close to P ∗ and because of thesymmetry of P ∗ , locally around t = 0, we expect to have oscillations of P ∗ around P so that eitherPlayer a or Player b can trade with the market maker. If P ∗ increases (resp. if P ∗ decreases) signifi-cantly beyond P , Player b (resp. Player a ) is likely to start an auction since his probability to tradewith the market maker in a short amount of time becomes smaller. Although he can trade, the otherplayer may also wish to trigger an auction, as his trading price becomes very unfavourable. So we seethat ad-hoc auctions ensure that the trading price does not deviate too much from the efficient price.The plots of V a and V b in Figure 3 are not even functions with respect to P − P ∗ . To explain it, takefor example the position of Player a . If P > P ∗ he can buy and launch an auction while if P < P ∗ he can only launch an auction. Consequently, the situation P > P ∗ is somehow more acceptablefor him. We recall that as Player a minimises and Player b maximises, the value functions aresymmetric with respect to the origin. We also see that for Player a , there is a large peak of thevalue function when P − P ∗ is slightly negative and only a small downward bump when it is slightlypositive. This means that for Player a , there is much more to lose when Player b can trade with themarket maker than to earn when he can trade with the market maker. This will be also confirmedin Table 3 below.Figure 3: Values of the game and average duration of the continuous phase, at time t = 0, asfunctions of P − P ∗ , with v a = v b = 0 . 1. 28 igure 4 : demand higher than supply for small investors. When v a = 0 . v b = 0 . b is better off than Player a . When P > P ∗ , Player a can trade with the market makerhence reducing the imbalance with respect to the volume of Player b . Player b will typically notimmediately trigger an auction because of the quite significant entry cost of the auction ˆ n = 3. Thisexplains the quite long duration of the auction phase in this situation and the downward peak ofthe value function of Player a . When P − P ∗ is negative, Player b can trade with the market makerwhich could lead to an even larger imbalance from Player a ’s perspective. Thus we expect Player a to trigger the auction in that case explaining the short length of the continuous phase and theflat behaviour of the value functions on the left of 0 (whatever P − P ∗ < 0, Player a will trigger anauction).Figure 4: Values of the game and average duration of the continuous phase, at time t = 0, asfunctions of P − P ∗ , with v a = 0 . v b = 0 . Figure 5 : supply higher than demand for large investors. When v a = 0 . v b = 0 . P > P ∗ , as previously, Player a can trade with the marketmaker, which improves even more its imbalance position with respect to the volume of Player b andit is particularly interesting when P is only slightly larger than P ∗ . In that case, Player b rapidlytriggers an auction to prevent Player a from trading. Note that contrary to the previous situation,the entry cost is not prohibitive here for Player b as v b = 0 . 15. When P is significantly larger than29 ∗ , the price becomes too bad for Player a who stops trading. Then a gaming situation occursbetween the two players explaining the delay before one of them triggers the auction. Regardingthe value functions, the peak of the orange graph is explained by the fact that it is very interestingfor Player b to trade with the market maker to reduce his imbalance with respect to the volume ofPlayer a (who may be reluctant to trigger an auction as v a is not very large).Figure 5: Values of the game and average duration of the continuous phase, at time t = 0, asfunctions of P − P ∗ , with v a = 0 . v b = 0 . We finally provide the value functions and average durations in the case of ad-hoc auctions, expensiveperiodic auctions (ˆ n = 3 and no trading allowed in the continuous phase), inexpensive periodicauctions (ˆ n = 1 and no trading allowed in the continuous phase) and CLOB. In the case of CLOB,the players trade only with the market maker and pay 1 /K for each trade. The average durationis then defined as the average time between two trades and the value as the amount paid per unitof time. The results are shown in Table 3.We notice first that, if continuous trading with the market maker is allowed, the average durationof the pre-auction phase is longer. This is because both players try to trade with the market makerif possible in order to push the settlement price of the next auction in their favour.30 a (1e-6) Average durationMarket design h = 20,ˆ n = 3 h = 20,ˆ n = 1 CLOB h = 20,ˆ n = 3 h = 20,ˆ n = 1 CLOBcontinuous trading allowed Yes No No No Yes No No No v a = 0 . v b = 0 . v a = 0 . v b = 0 . v a = 0 . v b = 0 . 05 7800.0 11606.7 10000.0 10000.0 33.3s 10.0s 0.0s 10.0s v a = 0 . v b = 0 . v a = 0 . v b = 0 . 15 -2841.8 0.0 0.0 10000.0 9.0s 5.0s 0.0s 10sTable 3: V a and average duration of the continuous trading phase for different values of v a and v b with q = 0 . v a = v b = 0 . 1, Player a prefers the case where there is no continuous trading. This is in agree-ment with our interpretation of Figure 3 since Player a has much more to lose when Player b cantrade with the market maker than to earn when he can trade with the market maker. Moreover, ifthe triggering volume is small (ˆ n = 1), the probability of the auctions to be balanced is large andthe player who cannot trade with the market maker triggers an auction quickly. The case ˆ n = 3provides an intermediary between periodic auctions and CLOB in terms of value functions.If v a and v b are small and asymmetric (either v a = 0 . 05 and v b = 0 . v a = 0 . v b = 0 . ad-hoc auctions. We explain this asfollows: if the larger player can trade with the market maker, he is able to liquidate his temporarysurplus at a low cost with the market maker and so suffers less from price impact in the auction,which is more balanced than in the situation without continuous trading. In this case, it is too costlyfor the smaller player to trigger an auction since ˆ n is too high compared to the target 0 . 05. Thelarger player is thus the first to trigger the auction if the price becomes too unfavourable, in a waysignalling to the smaller player that it is preferable to trade at the forthcoming auction instead ofat the clearing price. Otherwise, if the smaller player trades with the market maker during the con-tinuous trading phase, the larger player triggers the auction to protect himself from an excessivelyunfavourable price at the auction. The smaller player benefits from information leakage/marketimpact generated by the larger player, while the larger player uses his informational advantage ofbeing the larger player by capturing mistimed liquidity from the smaller player. In both cases, thelarger player is the one triggering the auction and benefits from the continuous trading phase. Thisis in agreement with Figure 4 where V a takes its lowest value for P > P ∗ with P close to P ∗ . Inaddition, compared to the case without market maker or with | P − P ∗ | large, the temporary targetimbalance has less impact on the distance between the clearing price and the efficient price. This isa direct consequence of the surplus of orders from the larger player being absorbed by the market31aker. We consider this an advantage of ad-hoc auctions: the clearing price has less volatility.We now turn to v a = 0 . v b = 0 . 15. As before, if Player a (smaller player) trades with themarket maker, he can liquidate part of his volume but Player b (larger player) quickly triggersan auction to prevent him from doing so. The larger player can indeed trigger the auction sinceˆ n = 3 coincides with his target. When Player b trades with the market maker, unlike the previouscase, the auction triggering cost is reasonable for Player a . The continuous phase appears as anopportunity for Player a to prevent Player b from mitigating his inventory since in this case Player a triggers the auction. This is in accordance with Figure 5 above. Conversely, we observe that thevalue functions of Player b are quite similar considering ad-hoc auctions or classical periodic auc-tions. One conclusion is that for large investors, the smaller one benefits a lot from ad-hoc auctionscompared to periodic auctions and CLOB, while the larger one is quite indifferent between ad-hoc and periodic auctions.The parameter q plays quite an important role since it dictates the probability of an auction to bebalanced out. We refer to Appendix E for the value functions and average durations with q = 0 . n = 3 since itmitigates price impact during the auction. The value functions are close to those observed withperiodic auctions but have the attractive property of having very long periods of continuous trading:the price remains constant for a long time while with periodic auctions, auctions are triggered assoon as someone needs to trade. Also the larger player still benefits a lot from being able to tradewith the market maker. Acknowledgments The authors gratefully acknowledge the financial support of the ERC Grant 679836 Staqamof andthe Chaire Analytics and Models for Regulation . They are also thankful to Alexandra Givry, IrisLucas and Eric Va for insightful discussions. 32 eferences A¨ıd, R., Basei, M., Callegaro, G., Campi, L., and Vargiolu, T. (2020). Nonzero-sum stochasticdifferential games with impulse controls: A verification theorem with applications. Math.Oper. Res. , 45:205–232.Almgren, R. and Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk ,3:5–40.Aquilina, M., Budish, E. B., and O’Neill, P. (2020). Quantifying the high-frequency trading “armsrace”: a simple new methodology and estimates. Chicago Booth Research Paper , 20(16).Basei, M., Cao, H., and Guo, X. (2019). Nonzero-sum stochastic games with impulse controls. arXiv:1901.08085 .Budish, E., Cramton, P., and Shim, J. (2015). The high-frequency trading arms race: frequent batchauctions as a market design response. The Quarterly Journal of Economics , 130(4):1547–1621.Carmona, R. and Delarue, F. (2018). Probabilistic Theory of Mean Field Games with ApplicationsI . Springer.Coquet, F. and Toldo, S. (2007). Convergence of values in optimal stopping and convergence ofoptimal stopping times. Electron. J. Probab. , 12:207–228.Cvitanic, J. and Karatzas, I. (1993). Hedging contingent claims with constrained portfolios. Ann.Appl. Probab. , 3(3):652–681.Delattre, S., Robert, C. Y., and Rosenbaum, M. (2013). Estimating the efficient price from theorder flow: a Brownian cox process approach. Stochastic Processes and their Applications ,123(7):2603–2619.Du, S. and Zhu, H. (2017). What is the optimal trading frequency in financial markets? The Reviewof Economic Studies , 84(4):1606–1651.El Euch, O., Mastrolia, T., Rosenbaum, M., and Touzi, N. (2018). Optimal make-take fees formarket making regulation. arXiv:1805.02741 .Farmer, D. and Skouras, S. (2012). Review of the benefits of a continuous market vs. randomisedstop auctions and of alternative priority rules (policy options 7 and 12). BIS. Business andmanagement . 33ricke, D. and Gerig, A. (2018). Too fast or too slow? Determining the optimal speed of financialmarkets. Quantitative Finance , 18(4):519–532.Garbade, K. and Silber, W. L. (1979). Structural organization of secondary markets: Clearingfrequency, dealer activity and liquidity risk. Journal of Finance , 34(3):577–93.Grigorova, M. and Quenez, M.-C. (2017). Optimal stopping and a non-zero-sum Dynkin game indiscrete time with risk measures induced by BSDEs. Stochastics , 89(1):259–279.Hamad`ene, S. and Mu, R. (2014). Bang–bang-type Nash equilibrium point for Markovian nonzero-sum stochastic differential game. Comptes Rendus Mathematiques , 352.Jacod, J. and Shiryaev, A. N. (1987). Limit theorems for stochastic processes , volume 288 of Grundlehren der mathematischen Wissenschaften . Springer-Verlag.Jusselin, P., Mastrolia, T., and Rosenbaum, M. (2019). Optimal auction duration: A price formationviewpoint. arXiv:1906.01713 .Laraki, R. and Solan, E. (2005). The value of zero-sum stopping games in continuous time. SIAMJ. Control and Optimization , 43:1913–1922.Laraki, R. and Solan, E. (2010). Equilibrium in two-player non-zero-sum Dynkin games in con-tinuous time. Stochastics An International Journal of Probability and Stochastic Processes ,85.Ludkovski, M. (2010). Stochastic switching games and duopolistic competition in emissions markets. SIAM Journal on Financial Mathematics , 2.Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the National Academy ofSciences , 36(1):48–49.Riedel, F. and Steg, J.-H. (2017). Subgame-perfect equilibria in stochastic timing games. Journalof Mathematical Economics , 72:36–50.Robert, C. and Rosenbaum, M. (2011). A new approach for the dynamics of ultra-high-frequencydata: The model with uncertainty zones. Journal of Financial Econometrics , 9(2):344–366.Shmaya, E. and Solan, E. (2014). Equivalence between random stopping times in continuous time. arXiv:1403.7886 .Solan, E., Tsirelson, B., and Vieille, N. (2012). Random stopping times in stopping problems andstopping games. arXiv:1211.5802 . 34toikov, S. (2018). The micro-price: a high-frequency estimator of future prices. QuantitativeFinance , 18(12):1959–1966.Touzi, N. (2013). Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE ,volume 29. Fields Institute Monographs.Touzi, N. and Vieille, N. (2002). Continuous-time dynkin games with mixed strategies. SIAMJournal on Control and Optimization , 41(4):1073–1088.Von Neumann, J. and Morgenstern, O. (1947). Theory of Games and Economic Behavior . PrincetonUniversity Press. 35 Proof of Proposition 3.1 The proof of the existence of an open-loop Nash equilibrium for the sub-game (7) is a direct exten-sion of the results of Hamad`ene and Mu (2014) and Jusselin et al. (2019), taking into considerationthe continuous trading phase, together with a smooth decomposition of the value function at theoptimum.We focus on the dynamic programming principle (8). We follow the same argument as in Cvitanicand Karatzas (1993) Proposition 6.2 or El Euch et al. (2018) Lemma A.4. First, let us write χ a := C aauc + N aτ,τ + h ( N aτ,τ + h − N bτ,τ + h ) K . From the definition of an OLNE, we have J a ( x , ( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) = inf λ a ∈U [0 ,τ ] E λ a ,λ b, ∗ (cid:104) L aτ + χ a τ + h (cid:105) . Using the tower property we get J a ( x , ( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) = inf λ a ∈U [0 ,τ ] E λ a ,λ b, ∗ (cid:104) L aτ + E λ a ,λ b, ∗ τ [ χ a ] τ + h (cid:105) . (11)Moreover applying Bayes’ formula leads to E λ a ,λ b, ∗ τ [ χ a ] = E λ ,λ τ (cid:2) Ψ λ a ,λ b, ∗ T + h Ψ λ a ,λ b, ∗ τ χ a (cid:3) = E λ a ,λ b, ∗ [ τ,T + h ] τ [ χ a ] ≥ essinf µ a ∈U [ τ,T + h ] E µ a ,λ b, ∗ [ τ,T + h ] τ [ χ a ] . (12)Therefore, using both (11) and (12) we obtain J a ( x , ( τ a, ∗ , λ a, ∗ ) , ( τ b, ∗ , λ b, ∗ )) ≥ inf λ a ∈U [0 ,τ ] E λ a ,λ b, ∗ (cid:104) L aτ + ξ aτ τ + h (cid:105) . (13)Conversely, let λ a ∈ U [0 ,τ ] and µ a ∈ U [ τ,T + h ] . Recalling the definition ( λ a ⊗ τ µ a ) u := λ au u ≤ τ + µ au τ λ − + 1 z < λ + + 1 z =0 (cid:15) a λ ∗ b ( z, (cid:15) b ) =1 z < λ − + 1 z > λ + + 1 z =0 (cid:15) b . As z λ ∗ a ( z, (cid:15) a ) and z λ ∗ b ( z, (cid:15) b ) do not depend on (cid:15) a and (cid:15) b , we simply denote them by z λ ∗ a ( z ) and z λ ∗ b ( z ). For any z, ˜ z ∈ R and any (cid:15) ∈ [ λ − , λ + ], we set H a, ∗ ( p ∗ , z, ˜ z, (cid:15) ) =1 p ∗ P z λ ∗ b (˜ z, (cid:15) ) H b, ∗ ( p ∗ , z, ˜ z, (cid:15) ) =1 p ∗ >P z λ ∗ b ( z ) + 1 p ∗ β j U } ,∂ Γ j ( U ) = { ( t, p ∗ , n a , n b , l a , l b ) ∈ [0 , T ) × R × N × N × R × R , G j = U } . together with its derivative operators D a U ( t, p ∗ , l a , l b , n a , n b ) = U ( t, p ∗ , l a + ( P − p ∗ ) , l b , n a + 1 , n b ) − U ( t, p ∗ , l a , l b , n a , n b ) D b U ( t, p ∗ , l a , l b , n a , n b ) = U ( t, p ∗ , l a , l b + ( p ∗ − P ) , n a , n b + 1) − U ( t, p ∗ , l a , l b , n a , n b ) DU = ( D a U, D b U ) T . The quantity D j U describes the change in the value of the process ( U ( t, P ∗ t , L at , L bt , N at , N bt )) t ∈ [0 ,T ] when Player j , j ∈ { a, b } , sends an order which triggers a trade at the fixed price P . The set Γ j ( U )is the domain on which Player j would rather have a game of value U than trigger an auction alone(and thus have a game of value G j ). The set ∂ Γ j ( U ) is the domain on which he is indifferent. Wehave the following result. Theorem B.1. Let V a and V b be two functions of ( t, p ∗ , l a , l b , n a , n b ) from [0 , T ] × R × R + × R + × N × N into R . Assume that there exist two maps ε a , ε b from R + × R × R + × R + × N × N into [ λ − , λ + ] such that(i) V a and V b are C in time on [0 , T ) and in their third (on R ) and fourth arguments (on R ), C in their second argument (on R ), and are solutions to the following variational system: max (cid:8) − L a ( t, p ∗ , n a , n b , ∂ t V a , ∂ pp V a , ∂ l a V a , ∂ l b V a , DV a , DV b , ε b ) , V a − G a (cid:9) = 0 , on Γ b ( V b ) l a + g second a ( t,n a ,n b ) t + h = V a on ∂ Γ b ( V b ) V a ( T, p ∗ , l a , l b , n a , n b ) = l a + g T a ( T,n a ,n b ) T + h min (cid:8) − L b ( t, p ∗ , n a , n b , ∂ t V b , ∂ pp V b , ∂ l a V b , ∂ l b V b , DV b , DV a , ε a ) , V b − G b (cid:9) = 0 , on Γ a ( V a ) − l b + g second b ( t,n a ,n b ) t + h = V b on ∂ Γ a ( V a ) V b ( T, p ∗ , l a , l b , n a , n b ) = − l b + g T b ( T,n a ,n b ) T + h . (18) (ii) g second a ≤ g sim a on ∂ Γ b ( V b ) and g second b ≥ g sim b on ∂ Γ a ( V a ) . hen (( τ a , λ ∗ a ( DV a , ε a ) ⊗ τ ˆ λ a ) , ( τ b , λ ∗ b ( DV b , ε b ) ⊗ τ ˆ λ b )) is an OLNE in the sense of Definition 3.1,where τ a = inf { t ≥ , ( t, P ∗ t , L at , L bt , N at , N bt ) ∈ ∂ Γ a ( V a ) } τ b = inf { t ≥ , ( t, P ∗ t , L at , L bt , N at , N bt ) ∈ ∂ Γ b ( V b ) } . Remark B.1. The differentiability conditions are very strong. In a non bang-bang case, extend-ing the results of A¨ıd et al. (2020) to the case of jump processes, it is possible to show that C -differentiability in the third and fourth arguments is enough if ∂ Γ a ( V a ) and ∂ Γ b ( V b ) are Lips-chitz surfaces. Nevertheless, note that in Theorem B.1, Condition ( ii ) is easier to meet when ˆ n = ˆ n ab = 0 . This is because the stopping domains are allowed to intersect since for i = a, b wehave g first i = g sim i = g second i . Remark B.2. Suppose that there exists a solution to System (18) and that the players play theassociated Nash equilibrium. Then, as soon as g first a (cid:54) = g second a and g first b (cid:54) = g second b , necessarily wemust have ∂ Γ a ( V a ) ∩ ∂ Γ b ( V b ) = ∅ , i.e. the two players never trigger an auction at the same time. In practice, we have g first a (cid:54) = g second a and g first b (cid:54) = g second b if ˆ n (cid:54) = 0 and h is not too large. Also, note that from our numerical investigations,it seems that there is no uniqueness for the solution of System (18) .Proof of the verification theorem B.1. Suppose that the functions V a and V b satisfy the above con-ditions with the maps ε a and ε b and that Player b plays the strategy ( τ b , λ ∗ b ( DV b , ε b ) ⊗ τ ˆ λ b ). UsingProposition 8, we see that it is optimal for Player a to play ˆ λ a after τ , thus obtaining ξ aτ at τ . For t ≤ τ b , we write V a, ∗ t = ess inf τ a, ∗ ∈T t,T ,λ a ∈U [ t,T + h ] E λ a ,λ b, ∗ t (cid:2) L aτ + ξ aτ τ + h (cid:3) . On ∂ Γ b ( V b ), as t ≤ τ b and because of the definition of τ b , we necessarily have t = τ b . So Player b triggers an auction, from ( ii ) V a, ∗ t = min( l a + g second a ( t, n a , n b ) t + h , l a + g sim a ( t, n a , n b ) t + h ) = l a + g second a ( t, n a , n b ) t + h and Player a does not trigger an auction. On Γ b ( V b ), necessarily t < τ b , i.e. Player b does nottrigger an auction. Player a then solves a classical optimal stopping problem. Standard dynamic Here ˆ λ a and ˆ λ b denote the strategies played by Player a and Player b during the auction given by Proposition3.1. (cid:8) − L a ( t, p ∗ , n a , n b , ∂ t V, ∂ pp V, ∂ l a V, ∂ l b V, DV, DV b , ε b ) , V − G a (cid:9) = 0for the value of Player a ’s game. We now detail the computation of the generator L a . As theproblem depends only on ( t, P ∗ t , N at , N bt , L at , L bt ), Itˆo’s formula provides the following expression ∂ t V + 12 σ ∂ pp V + q (cid:88) i ∈{ a,b } ( v i t − N it ) ∂ l i V + inf λ a ∈ [ λ − ,λ + ] λ a P >P ∗ t D a V + λ ∗ b ( DV b , ε b ) P P ∗ t D a V + λ ∗ b ( DV b , ε b ) P
0, ( t, P ∗ t , L at , L bt , N at , N bt ) ∈ ∂ Γ a ( V a ) } . C Proofs of Section 3.2 C.1 Proof of Theorem 3.1 We prove by backward induction on k ∈ { Tδ , ..., } that U ak = ess inf τ a ∈T dkδ,T ,λ a ∈U [ kδ,T + h ] E λ a , ˜ λ b, ∗ k kδ (cid:2) L aτ + C aauc + Naτ,τ + h ( Naτ,τ + h − Nbτ,τ + h ) K τ + h (cid:3) U bk = ess sup τ b ∈T dkδ,T ,λ b ∈U [ kδ,T + h ] E ˜ λ a, ∗ k ,λ b kδ (cid:2) − L bτ −C bauc + Nbτ,τ + h ( Naτ,τ + h − Nbτ,τ + h ) K τ + h (cid:3) τ a = τ b = τ ∗ k , λ a = ˜ λ a, ∗ k and λ b = ˜ λ b, ∗ k with τ ∗ k = δ inf (cid:110) l ∈ (cid:74) k, T /δ (cid:75) , U al = L alδ + g first a ( lδ, N alδ , N blδ ) lδ + h or U bl = − L blδ + g first b ( lδ, N alδ , N blδ ) lδ + h (cid:111) , ˜ λ a, ∗ k = (cid:0) (cid:78) l ∈ (cid:74) k,T/δ − (cid:75) λ a, ∗ l (cid:1) ⊗ τ ∗ ˆ λ a and ˜ λ b, ∗ k = (cid:0) (cid:78) l ∈ (cid:74) k,T/δ − (cid:75) λ b, ∗ l (cid:1) ⊗ τ ∗ ˆ λ b . Applying this result at k = 0we will get that ( τ ∗ , ˜ λ a, ∗ ) , ( τ ∗ , ˜ λ b, ∗ ) is an OLNED. For k = T /δ , the result follows directly fromProposition 3.1. Suppose the result holds for k + 1 ∈ (cid:74) , T /δ (cid:75) . We show that it holds for k . Wethus assume that U ak +1 = ess inf τ a ∈T d ( k +1) δ,T ,λ a ∈U [( k +1) δ,T + h ] E λ a , ˜ λ b, ∗ k +1 ( k +1) δ (cid:2) L aτ + C aauc + Naτ,τ + h ( Naτ,τ + h − Nbτ,τ + h ) K τ + h (cid:3) U bk +1 = ess sup τ b ∈T d ( k +1) δ,T ,λ b ∈U [( k +1) δ,T + h ] E ˜ λ a, ∗ k +1 ,λ b ( k +1) δ (cid:2) − L bτ −C bauc + Nbτ,τ + h ( Naτ,τ + h − Nbτ,τ + h ) K τ + h (cid:3) . From the definition of λ a, ∗ k together with our induction assumption above, we have E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 ]= ess inf λ ak ∈U [ kδ, ( k +1) δ ] E λ ak ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) = ess inf λ ak ∈U [ kδ, ( k +1) δ ] E λ ak ,λ b, ∗ k kδ (cid:104) ess inf τ a ∈T d ( k +1) δ,T ,λ ak +1 ∈U [( k +1) δ,T + h ] E λ ak +1 , ˜ λ b, ∗ k +1 ( k +1) δ (cid:2) L aτ + C aauc + N aτ,τ + h ( N aτ,τ + h − N bτ,τ + h ) K τ + h (cid:3)(cid:105) . (19)We aim at showing thatess inf λ ak ∈U [ kδ, ( k +1) δ ] E λ ak ,λ b, ∗ k kδ (cid:104) ess inf τ a ∈T ( k +1) δ,T ,λ ak +1 ∈U [( k +1) δ,T + h ] E λ ak +1 , ˜ λ b, ∗ k +1 ( k +1) δ (cid:2) L aτ + C aauc + N aτ,τ + h ( N aτ,τ + h − N bτ,τ + h ) K τ + h (cid:3)(cid:105) = ess inf τ a ∈T ( k +1) δ,T ,λ ak ∈U [ kδ,T + h ] E λ ak , ˜ λ b, ∗ k kδ (cid:2) L aτ + C aauc + N aτ,τ + h ( N aτ,τ + h − N bτ,τ + h ) K τ + h (cid:3) , (20)where abusing notation slightly we write τ = T ∧ τ a ∧ τ ∗ k +1 . Let χ aT ∧ τ a ∧ τ ∗ k +1 = L aτ + C aauc + N aτ,τ + h ( N aτ,τ + h − N bτ,τ + h ) K τ + h and τ a be in T d ( k +1) δ,T . Using the tower property we get42 λ ak , ˜ λ b, ∗ k kδ (cid:2) χ aT ∧ τ a ∧ τ ∗ k +1 (cid:3) = E λ ak , ˜ λ b, ∗ k kδ (cid:104) E λ ak , ˜ λ b, ∗ k ( k +1) δ (cid:2) χ aT ∧ τ a ∧ τ ∗ k +1 (cid:3)(cid:105) = E λ ak , ˜ λ b, ∗ k kδ (cid:104) E λ ak, [( k +1) δ,T ] , ˜ λ b, ∗ k +1 ( k +1) δ (cid:2) χ aT ∧ τ a ∧ τ ∗ k +1 (cid:3)(cid:105) = E λ ak, [ kδ, ( k +1) δ ] ,λ b, ∗ k kδ (cid:104) E λ ak, [( k +1) δ,T ] , ˜ λ b, ∗ k +1 ( k +1) δ (cid:2) χ aT ∧ τ a ∧ τ ∗ k +1 (cid:3)(cid:105) ≥ E λ ak, [ kδ, ( k +1) δ ] ,λ b, ∗ k kδ (cid:104) ess inf τ a ∈T d ( k +1) δ,T ,λ ak +1 ∈U [( k +1) δ,T + h ] E λ ak +1 , ˜ λ b, ∗ k +1 ( k +1) δ (cid:2) χ aT ∧ τ a ∧ τ ∗ k +1 (cid:3)(cid:105) ≥ ess inf λ ak ∈U d [ kδ, ( k +1) δ ] E λ ak ,λ b, ∗ k kδ (cid:104) ess inf τ a ∈T d ( k +1) δ,T ,λ ak +1 ∈U [( k +1) δ,T + h ] E λ ak +1 , ˜ λ b, ∗ k +1 ( k +1) δ (cid:2) χ aT ∧ τ a ∧ τ ∗ k +1 (cid:3)(cid:105) , which gives one inequality in (20) by taking the essential infimum over τ a ∈ T d ( k +1) δ,T and λ ak ∈U [ kδ,T + h ] . We now turn to the other inequality. Let λ a ∈ U [ kδ, ( k +1) δ ] and µ a ∈ U [( k +1) δ,T + h ] . We recallthe definition ( λ a ⊗ ( k +1) δ µ a ) u := λ au u ≤ ( k +1) δ + µ au ( k +1) δ − L bkδ + g first b ( kδ, N akδ , N bkδ ) kδ + h , the extrema are not reached at kδ , i.e. the optimal stopping times are still equal to τ ∗ k +1 and thetrading rates are given by λ a, ∗ k ⊗ ( k +1) δ λ a, ∗ k +1 = ˜ λ a, ∗ k and λ b, ∗ k ⊗ ( k +1) δ λ b, ∗ k +1 = ˜ λ b, ∗ k . Else at least oneplayer triggers an auction and the optimal stopping times are equal to kδ . Consequently, τ ∗ k = δ inf (cid:110) l ∈ (cid:74) k, T /δ (cid:75) , U al = L alδ + g first a ( lδ, N alδ , N blδ ) lδ + h or U bl = − L blδ + g first b ( lδ, N alδ , N blδ ) lδ + h (cid:111) . .2 Proof of Theorem 3.3 Step 1: Construction of a (good) stopping strategy. Fix τ b ∈ T ,T and λ b ∈ U . Let ε > λ ∗ ∈ U and τ ∗ ∈ T ,T such that E λ ∗ ,λ b (cid:2) L aτ ∗ ∧ τ b + ξ aτ ∗ ∧ τ b τ ∗ ∧ τ b + h (cid:3) ≤ inf τ ∈T ,T ,λ ∈U E λ,λ b (cid:2) L aτ ∧ τ b + ξ aτ ∧ τ b τ ∧ τ b + h (cid:3) + ε and define τ ∗ d = δ (cid:100) τ ∗ δ (cid:101) τ ∗ <τ b + τ ∗ ≥ τ b τ ∗ . Also let λ a = λ ∗ ⊗ τ ∗ ˆ λ − where ˆ λ − ∈ U is the constantstrategy equal to λ − . Step 2: Comparison with the optimal payoff. We decompose the error made by choosing τ ∗ d insteadof τ ∗ into three terms:0 ≤ E λ a ,λ b (cid:2) L aτ ∗ d ∧ τ b + ξ aτ ∗ d ∧ τ b τ ∗ d ∧ τ b + h (cid:3) − inf τ ∈T ,T ,λ ∈U E λ,λ b (cid:2) L aτ ∧ τ b + ξ aτ ∧ τ b τ ∧ τ b + h (cid:3) ≤ E λ a ,λ b (cid:2) L aτ ∗ d ∧ τ b + ξ aτ ∗ d ∧ τ b τ ∗ d ∧ τ b + h (cid:3) − E λ ∗ ,λ b (cid:2) L aτ ∗ ∧ τ b + ξ aτ ∗ ∧ τ b τ ∗ ∧ τ b + h (cid:3) + ε = E λ a ,λ b (cid:2) L aτ ∗ d ∧ τ b + ξ aτ ∗ d ∧ τ b τ ∗ d ∧ τ b + h − L aτ ∗ ∧ τ b + ξ aτ ∗ ∧ τ b τ ∗ ∧ τ b + h (cid:3) + ε = E λ a ,λ b (cid:2) τ ∗ <τ b ( L aτ ∗ d ∧ τ b + ξ aτ ∗ d ∧ τ b τ ∗ d ∧ τ b + h − L aτ ∗ ∧ τ b + ξ aτ ∗ ∧ τ b τ ∗ ∧ τ b + h ) (cid:3) + ε = E λ a ,λ b (cid:2) τ ∗ <τ b ( L aτ ∗ d ∧ τ b + ξ aτ ∗ d ∧ τ b τ ∗ d ∧ τ b + h − L aτ ∗ ∧ τ b + g first a ( τ ∗ ∧ τ b ) τ ∗ ∧ τ b + h ) (cid:3) + ε = E λ a ,λ b (cid:2) τ ∗ <τ b ( L aτ ∗ d ∧ τ b + g first a ( τ ∗ d ∧ τ b , N aτ ∗ d ∧ τ b , N bτ ∗ d ∧ τ b ) τ ∗ d ∧ τ b + h − L aτ ∗ ∧ τ b + g first a ( τ ∗ ∧ τ b , N aτ ∗ ∧ τ b , N bτ ∗ ∧ τ b ) τ ∗ ∧ τ b + h ) (cid:3) + E λ a ,λ b (cid:2) τ ∗ <τ b ( L aτ ∗ d ∧ τ b + ξ aτ ∗ d ∧ τ b τ ∗ d ∧ τ b + h − L aτ ∗ d ∧ τ b + g first a ( τ ∗ d ∧ τ b , N aτ ∗ d ∧ τ b , N bτ ∗ d ∧ τ b ) τ ∗ d ∧ τ b + h ) (cid:3) + ε. We have τ ∗ ,d − τ ∗ → δ → v a τ and v b τ are replaced by the nearest integer andusing that v a δ ∈ N and v b δ ∈ N , we get g first a ( τ ∗ d ∧ τ b , N aτ ∗ d ∧ τ b , N bτ ∗ d ∧ τ b ) = g first a ( τ ∗ ∧ τ b , N aτ ∗ d ∧ τ b , N bτ ∗ d ∧ τ b ) . So, using that the intensities of the Poisson processes are bounded, we have E λ a ,λ b (cid:2) τ ∗ <τ b ( L aτ ∗ d ∧ τ b + g first a ( τ ∗ d ∧ τ b , N aτ ∗ d ∧ τ b , N bτ ∗ d ∧ τ b ) τ ∗ d ∧ τ b + h − L aτ ∗ ∧ τ b + g first a ( τ ∗ ∧ τ b , N aτ ∗ ∧ τ b , N bτ ∗ ∧ τ b ) τ ∗ ∧ τ b + h ) (cid:3) → δ → , τ ∗ , τ b , λ a and λ b . Also, E λ a ,λ b (cid:2) τ ∗ <τ b ( L aτ ∗ d ∧ τ b + ξ aτ ∗ d ∧ τ b τ ∗ d ∧ τ b + h − L aτ ∗ d ∧ τ b + g first a ( τ ∗ d ∧ τ b , N aτ ∗ d ∧ τ b , N bτ ∗ d ∧ τ b ) τ ∗ d ∧ τ b + h ) (cid:3) ≤ h E λ a ,λ b (cid:2) max(0 , ξ aτ ∗ d ∧ τ b − g first a ( τ ∗ d ∧ τ b , N aτ ∗ d ∧ τ b , N bτ ∗ d ∧ τ b )) (cid:3) ≤ h sup λ a ∈U ,λ b ∈U E λ a ,λ b (cid:2) sup [0 ,T ] (max(max( g sim a ( t, N at , N bt ) , g second a ( t, N at , N bt )) − g first a ( t, N at , N bt ) , (cid:3) = ε a . Step 3: Conclusion. We have shown that for any ε > 0, there exists ˆ δ a > δ ≤ ˆ δ a ,then for any τ b and λ b chosen by Player b , we can find some τ ∗ d ∈ T ,T and λ a ∈ U such that0 ≤ E λ a ,λ b (cid:2) L aτ ∗ d ∧ τ b + ξ aτ ∗ d ∧ τ b τ ∗ d ∧ τ b + h (cid:3) − inf τ ∈T ,T ,λ ∈U E λ,λ b (cid:2) L aτ ∧ τ b + ξ aτ ∧ τ b τ ∧ τ b + h (cid:3) ≤ ε + ε a . Now let (( λ a , λ b ) , ( τ a , τ b )) be an OLNED for some δ < ˆ δ a and let τ ∗ d ∈ T ,T and λ a ∈ U as in Step1 . Remark that τ ∗ d ∈ T d ,T and so0 ≤ E λ a ,λ b (cid:2) L aτ a ∧ τ b + ξ aτ a ∧ τ b τ a ∧ τ b + h (cid:3) − inf τ ∈T ,T ,λ ∈U E λ,λ b (cid:2) L aτ ∧ τ b + ξ aτ ∧ τ b τ ∧ τ b + h (cid:3) = inf τ ∈T d ,T ,λ ∈U E λ,λ b (cid:2) L aτ ∧ τ b + ξ aτ ∧ τ b τ ∧ τ b + h (cid:3) − inf τ ∈T ,T ,λ ∈U E λ,λ b (cid:2) L aτ ∧ τ b + ξ aτ ∧ τ b τ ∧ τ b + h (cid:3) ≤ E λ a ,λ b (cid:2) L aτ ∗ d ∧ τ b + ξ aτ ∗ d ∧ τ b τ ∗ d ∧ τ b + h (cid:3) − inf τ ∈T ,T ,λ ∈U E λ,λ b (cid:2) L aτ ∧ τ b + ξ aτ ∧ τ b τ ∧ τ b + h (cid:3) ≤ ε + ε a . Similarly, we find ˆ δ b > λ a , λ b ) , ( τ a , τ b )) is an OLNED for some δ < ˆ δ b , then0 ≤ sup τ ∈T ,T ,λ ∈U E λ a ,λ (cid:2) − L bτ a ∧ τ + ξ bτ a ∧ τ τ a ∧ τ + h (cid:3) − E λ a ,λ b (cid:2) − L bτ a ∧ τ b + ξ bτ a ∧ τ b τ a ∧ τ b + h (cid:3) ≤ ε + ε b . Algorithms D.1 Value functions when ˆ n = ˆ n ab = 0 Algorithm 1: Computation of the value functions in the discretised game Result: Value functions and probability of triggering an auctionSet ( U aT/δ , U bT/δ ) = ( L aT + g T a ( T, P ∗ T , N aT , N bT ) T + h , − L bT + g T b ( T, P ∗ T , N aT , N bT ) T + h ) for k ∈ { T /δ − , ..., } do Let ( λ a, ∗ k , λ bk ) ∈ U k such that E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) = ess inf λ a ∈U [ kδ, ( k +1) δ ] E λ a ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3) = ess sup λ b ∈U [ kδ, ( k +1) δ ] E λ a, ∗ k ,λ b kδ (cid:2) U bk +1 (cid:3) , if E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) ≤ L akδ + g first a ( kδ,P ∗ kδ ,N akδ ,N bkδ ) kδ + h and E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3) ≥ − L bkδ + g first b ( kδ,P ∗ kδ ,N akδ ,N bkδ ) kδ + h then Set ( U ak , U bk ) = ( E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) , E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3) ) else Set ( U ak , U bk ) = ( L akδ + g first a ( kδ,P ∗ kδ ,N akδ ,N bkδ ) kδ + h , − L bkδ + g first b ( kδ,P ∗ kδ ,N akδ ,N bkδ ) kδ + h ) endend .2 Value functions in the general case: randomised discrete stoppingtime Algorithm 2: Computation of the value functions in the discretised game Result: Value functions and probability of triggering an auctionSet ( U aT/δ , U bT/δ ) = ( L aT + g T a ( T, P ∗ T , N aT , N bT ) T + h , − L bT + g T b ( T, P ∗ T , N aT , N bT ) T + h ) for k ∈ { T /δ − , ..., } do Let • ( λ a, ∗ k , λ b, ∗ k ) ∈ U k such that E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) = ess inf λ a ∈U [ kδ, ( k +1) δ ] E λ a ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3) = ess sup λ b ∈U [ kδ, ( k +1) δ ] E λ a, ∗ k ,λ b kδ (cid:2) U bk +1 (cid:3) , • ( p ak , p bk ) ∈ [0 , such that ( p ak , − p ak ) and ( p bk , − p bk ) define the mixed strategiesof a Nash equilibrium for the discrete game of Table 2.Set U ak = p ak p bk L akδ + g sim a kδ + h + p ak (1 − p bk ) L akδ + g first a kδ + h + (1 − p ak ) p bk L akδ + g second a kδ + h + (1 − p ak )(1 − p bk ) E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U ak +1 (cid:3) U bk = p ak p bk − L bkδ + g sim b kδ + h + p ak (1 − p bk ) − L bkδ + g second b kδ + h + (1 − p ak ) p bk − L bkδ + g first b kδ + h + (1 − p ak )(1 − p bk ) E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) U bk +1 (cid:3) where the arguments of the functions g first a , g second a , g sim a , g first b , g second b and g sim b are( kδ, P ∗ kδ , N akδ , N bkδ ). end D.3 Average duration of the continuous trading phase The average duration is defined by E k = E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) τ − kδ | τ ≥ kδ (cid:3) for k ∈ { , ..., T /δ } . In particular E = E λ a, ∗ ,λ b, ∗ [ τ ]. Assuming that it is a Markovian function of thestate variables P ∗ , N a , N b , L a , L b and that the PDE obtained from the Feyman-Kac formula has a48nique solution, we compute E k with the following algorithm. Algorithm 3: Computation of the average duration of the continuous phase in the discre-tised game Result: Average duration of the continuous phaseSet E T/δ = 0 for k ∈ { T /δ − , ..., } do Compute E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) E k +1 (cid:3) with the Feyman-Kac formula: E λ a, ∗ k ,λ b, ∗ k kδ (cid:2) E k +1 (cid:3) = e kδ where e is the solution of the equation e ( k +1) δ = E k +1 ∂ t e + σ ∂ pp e + 1 + q ( v a t − n a ) ∂ l a e + q ( v b t − n b ) ∂ l b e + P >p λ a, ∗ ,kt D a e + P
05 14196.4 22874.1 12000.5 10000.0 56.9s 14.9s 0.0s 20.0s v a = 0 . v b = 0 . v a = 0 . v b = 0 . 15 16450.0 16309.9 9510.7 10000.0 27.1s 3.9s 0.0s 10.0sTable 4: V a and average duration of the continuous trading phase for different values of v a and v b with q = 0 ..