A simple case of rationality of escalation
aa r X i v : . [ c s . G T ] J un A simple case of rationality of escalation
Pierre Lescanne
University of Lyon, ´Ecole normale sup´erieure de Lyon, CNRS (LIP),46 all´ee d’Italie, 69364 Lyon, France
Abstract.
Escalation is the fact that in a game (for instance an auc-tion), the agents play forever. It is not necessary to consider complexexamples to establish its rationality. In particular, the 0 , Keywords: economic game, infinite game, sequential game, crash, esca-lation, speculative bubble, coinduction, auction. [T]he future of economics is increasingly technicalwork that is founded on the vision that the economyis a complex system.
David Collander [6]Sequential games are the natural framework for decision processes. In thispaper we study a decision phenomenon called escalation . Finite sequential games(also known as extensive games) have been introduced by Kuhn [9] and subgameperfect equilibria have been introduced by Selten [19] whereas escalation has beenintroduced by Shubik [20]. Sequential games are games in which each player playsone after the other (or possibly after herself). In some specific infinite games,it has been showed that escalation cannot occur among rational players. Herewe show on a simple example, the 0,1 game, that this is not the case if oneuses coinduction. In addition the 0,1 game has nice properties which make it anexcellent paradigm of escalation and a good domain of application for coalgebrasand coinduction.
That “rational agents” should not engage in such [es-calation] behavior seems obvious.
Wolfgang Leininger [11]Escalation in sequential games is a classic of game theory and it is admittedthat escalation is irrational. The rationality which we consider is that giveny equilibria. It has been proved that in finite sequential games, rationality isobtained by a specific equilibrium called backward induction (see Appendix).More precisely a consequence of Aumann’s theorem [2] says that an agent takesa rational decision in a finite sequential game if she makes her choice accordingto backward induction. In this paper we generalize backward induction into subgame perfect equilibria and we consider naturally that rationality is reachedby subgame perfect equilibria (
SPE in short) relying on Capretta’s [5] extensionof Aumann’s theorem.
What is escalation?
In a sequential game, escalation is the possibility that agentstake rational decisions forever without stopping. This phenomenon has beenevidenced by Shubik [20] in a game called the dollar auction . Without beingvery difficult, its analysis is relatively involved, because it requires infinitelymany strategy profiles indexed by n ∈ N [12]. Moreover in each step there aretwo and only two equilibria. By an observation of the past decisions of heropponent an agent could get a clue of her strategy and might this way avoidescalation. This blindness of the agents is perhaps not completely realistic andwas criticized (see [14] Section 4.2). In this paper, we propose an example whichis much simpler theoretically and which offers infinitely many equilibria at eachstep. Due to the form of the equilibria, the agent has no clue on which strategyis taken by her opponent. Escalation and infinite games.
Books and articles [7,8,17,11,16] which cover esca-lation take for granted that escalation is irrational. Following Shubik, all acceptthat escalation takes place and can only take place in an infinite game, but theirargument uses a reasoning on finite games. Indeed, if one cuts the infinite gamein which escalation is supposed to take place at a finite position, one gets a finitegame, in which the only right and rational decision is to never start the game,because the only backward induction equilibrium corresponds to not start play-ing. Then the result is extrapolated by the authors to infinite games by makingthe size of the game to grow to infinity. However, it has been known for a longtime at least since Weierstraß [24], that the “cut and extrapolate” method iswrong (see Appendix). For Weierstraß this would lead to the conclusion that theinfinite sum of differentiable functions would be differentiable whereas he hasexhibited a famous counterexample. In the case of infinite structures like infi-nite games, the right reasoning is coinduction. With coinduction we were ableto show that the dollar auction has a rational escalation [15,14]. Currently, sincethe tools used generally in economics are pre-coinduction based, they concludethat bubbles and crises are impossible and everybody’s experience has witnessedthe opposite. Careful analysis done by quantitative economists, like for instanceBouchaud [3,4], have shown that bursts, which share much similarities with esca-lation, actually take place at any time scale. Escalation is therefore an intrinsicfeature of economics. Consequently, coinduction is the tool that economists whocall for a refoundation of economics [6,3] are waiting for [25]. tructure of the paper
This paper is structured as follows. In Section 2 we presentinfinite games, infinite strategy profiles and infinite strategies, then we describethe 0,1-game in Section 3. Last, we introduce the concept of equilibrium (Sec-tions 4 and 5) and we discuss escalation (Section 6). In an appendix, we talkabout finite games and finite strategy profiles.
Our aim is not to present a general theory. For this the reader is invited to lookat [1,14,15]. But we want to give a taste of infinite sequential games througha very simple one. This game has two agents and two choices. To support ourclaim about the rationality of escalation, we do not need more features. In [14],we have shown the existence of a big conceptual gap between finite games andinfinite games.Assume that the set P of agents is made of two agents called A and B . In thisframework, an infinite sequential two choice game has two shapes. In the firstshape, it is an ending position in which case it boils down to the distributionof the payoffs to the agents. In other words the game is reduced to a function f : A f A , B f B and we write it h f i . In the second shape, it is a genericgame with a set C made of two potential choices: d or r ( d for down and r for right) . Since the game is potentially infinite, it may continue forever. Thusformally in this most general configuration a game can be seen as a triple: g = h p, g d , g r i . where p is an agent and g d and g r are themselves games. The subgame g d cor-responds to the down choice, i.e., the choice corresponding to go down and thesubgame g r corresponds to the right choice, i.e., the choices corresponding to goto the right. In other words, we define a functor: h i : X → Payoff + P × X × X . of which Game is the final coalgebra and where P = { A , B } and Payoff = R P . From a game, one can deduce strategy profiles (later we will also say simply profiles ), which is obtained by adding a label, at each node, which is a choicemade by the agent. A choice belong to the set { d, r } . In other words, a strategyprofile is obtained from a game by adding, at each node, a new label, namelya choice. Therefore a strategy profile which does not correspond to an endinggame is a quadruple: s = hh p, c, s d , s r ii , where p is an agent ( A or B ), c is choice ( d or r ), and, s d and s r are two strategyprofiles. The strategy profile which corresponds to an ending position has nohoice, namely it is reduced to a function hh f ii = hh A f A , B f B ii . From astrategy profile, one can build a game by removing the choices: game ( hh f ii ) = h f i game ( hh p, c, s d , s r ii ) = h p, game ( s d ) , game ( s r ) i game ( s ) is the underlying game of the strategy profile s .Given a strategy profile s , one can associate, by induction, a (partial) payofffunction b s , as follows: when s = hh f ii b s = f when s = hh p, d, s d , s r ii b s = b s d when s = hh p, r, s d , s r ii b s = b s r b s is not defined if its definition runs in an infinite process. For instance, if s A , ∞ isthe strategy profile defined in Section 6, [ s A , ∞ is not defined. To ensure that weconsider only strategy profiles where the payoff function is defined we restrict tostrategy profiles that are called convergent , written s ↓ (or sometimes prefixed ↓ ( s )) and defined as the least predicate satisfying s = hh f ii ∨ s = hh p, d, s d , s r ii ⇒ s d ↓ ∧ hh p, r, s d , s r ii ⇒ s r ↓ . Proposition 1. If s ↓ , then b s is defined.Proof. By induction. If s = hh f ii , then since b s = f and f is defined, b s is defined.If s = hh p, c, s d , s r ii , there are two cases: c = d or c = r . Let us look at c = d .If c = d , b s d is defined by induction and since b s = b s d , we conclude that b s isdefined.The case c = r is similar.As we will consider the payoff function also for subprofiles, we want the pay-off function to be defined on subprofiles as well. Therefore we define a propertystronger than convergence which we call strong convergence . We say that a strat-egy profile s is strongly convergent and we write it s ⇓ if it is the largest predicatefulfilling the following conditions. – hh p, c, s d , s r ii ⇓ if • hh p, c, s d , s r ii is convergent, • s d is strongly convergent, • s r is strongly convergent. – hh f ii is always strongly convergentMore formally: s ⇓ k s c , + s = hh f ii ∨ ( s = hh p, c, s d , s r ii ∧ s ↓ ∧ s d ⇓ ∧ s r ⇓ ) . here is however a difference between the definitions of ↓ and ⇓ . Wherever s ↓ is defined by induction , from the ending games to the game, s ⇓ is defined by coinduction .Both concepts are based on the fixed-point theorem established by Tarski [21].The definition of ⇓ is typical of infinite games and means that ⇓ is invariant alongthe infinite game. To make the difference clear between the definitions, we usethe symbol k s i , + for inductive definitions and the symbol k s c , + for coinductivedefinitions. By the way, the definition of the function game is also coinductive.We can define the notion of subprofile , written - : s ′ - s k s i , + s ′ ∼ s s ∨ s = hh p, c, s d , s r ii ∧ ( s ′ - s d ∨ s ′ - s r ) , where ∼ s is the bisimilarity among profiles defined as the largest binary predicate s ′ ∼ s s such that s ′ = hh f ii = s ∨ ( s ′ = hh p, c, s ′ d , s ′ r ii ∧ s = hh p, c, s d , s r ii ∧ s ′ d ∼ s s d ∧ s ′ r ∼ s s r ) . Notice that since we work with infinite objects, we may have s s s ′ and s - s ′ - s . In other words, an infinite profile can be a strict subprofile of it-self. This is the case for s , ,a and s , ,b in Section 4. If a profile is stronglyconvergent, then the payoffs associated with all its subprofiles are defined. Proposition 2. If s ⇓ and if s - s , then b s is defined. We notice that ↓ characterizes a profile by a property of the head node, we wouldsay that this property is local. ⇓ is obtained by distributing the property alongthe game. In other words we transform the predicate ↓ and such a predicatetransformer is called a modality . Here we are interested by the modality always ,also written (cid:3) .Given a predicate Φ on strategy profiles, the predicate (cid:3) P is defined coin-ductively as follows: (cid:3) Φ ( s ) k s c , + Φ ( s ) ∧ s = hh p, c, s d , s r ii ⇒ ( (cid:3) Φ ( s d ) ∧ (cid:3) Φ ( s r )) . The predicate “is strongly convergent” is the same as the predicate “is alwaysconvergent”.
Proposition 3. s ⇓ ⇔ (cid:3) ↓ ( s ) . Roughly speaking a definition by induction works from the basic elements, here theending games, to constructed elements. Roughly speaking a definition by coinduction works on infinite objects, like an in-variant. .3 Strategies
The coalgebra of strategies is defined by the functor J K : X → R P + ( P + Choice ) × X × X where Choice = { d, r } . A strategy of agent p is a game in which some occurrencesof p are replaced by choices. A strategy is written J f K or J x, s , s K . By replacingthe choice made by agent p by the agent p herself, we can associate a game witha pair consisting of a strategy and an agent: st2g ( J f K , p ) = h f i st2g ( J x, st , st K , p ) = if x ∈ P then h x, st2g ( st , p ) , st2g ( st , p ) i else h p, st2g ( st , p ) , st2g ( st , p ) i . If a strategy st is really the strategy of agent p it should contain nowhere p and should contain a choice c instead. In this case we say that st is full for p and we write it st . p . J f K . p J x, st , st K . p k s c , + ( x / ∈ Choice ⇒ x = p ) ∧ st . p ∧ st . p . We can sum strategies to make a strategy profile. But for that we have to assumethat all strategies are full and underlie the same game. In other words, ( st p ) p ∈ P is a family of strategies such that: – ∀ p ∈ P , st p . p , – there exists a game g such that ∀ p ∈ P , st2g ( st p ) = g .We define M p ∈ P st p as follows: M p ∈ P J f K = hh f ii J c, st p ′ , , st p ′ , K ⊕ M p ∈ P \ p ′ J p ′ , st p, , st p, K = hh p ′ , c, M p ∈ P st p, , M p ∈ P st p, ii . We can show that the game underlying all the strategies is the game underlyingthe strategy profile which is the sum of the strategies.
Proposition 4. st2g ( st p ′ , p ′ ) = game ( M p ∈ P st p ) . A strategy is not the same as a strategy profile, which is obtained as the sum ofstrategies.
Comb games and the 0,1-game
We will restrict to simple games which have the shape of combs, ?>=<89:; A r * * d (cid:9) (cid:9) ?>=<89:; B r * * d (cid:9) (cid:9) ?>=<89:; A r * * d (cid:9) (cid:9) ?>=<89:; B r * * d (cid:9) (cid:9) ?>=<89:; A r * * d (cid:9) (cid:9) ?>=<89:; B r & & d (cid:9) (cid:9) r $ $ d (cid:6) (cid:6) r $ $ f f f f f f At each step the agents have only two choices, namely to stop or to continue.Let us call such a game, a comb game .We introduce infinite games by means of equations. Let us see how thisapplies to define the 0 , f , = A , B f , = A , B G , = h A , h f , i , G , i G , = h B , h f , i , G , i This means that we define an infinite sequential game G , in which agent A isthe first player and which has two subgames: the trivial game h f , i and the game G , defined in the other equation. The game G , can be pictured as follows: ?>=<89:; A r * * d (cid:9) (cid:9) ?>=<89:; B r * * d (cid:9) (cid:9) ?>=<89:; A r * * d (cid:9) (cid:9) ?>=<89:; B r * * d (cid:9) (cid:9) ?>=<89:; A r * * d (cid:9) (cid:9) ?>=<89:; B r & & d (cid:9) (cid:9) r $ $ d (cid:6) (cid:6) r $ $ , , , , , , S ( s ) k s c , + s = hh A , c, f , , s ′ ii ∧ S ( s ′ ) S ( s ) k s c , + s = hh B , c, f , , s ′ ii ∧ S ( s ′ ) . Notice that the 0 , , / ?>=<89:; A r % % d (cid:9) (cid:9) ?>=<89:; B r e e d (cid:9) (cid:9) , , / / ?>=<89:; A r ! ) d (cid:9) (cid:9) ?>=<89:; B r e e d (cid:5) (cid:13) , , / / ?>=<89:; A r % % d (cid:5) (cid:13) ?>=<89:; B r a i d (cid:9) (cid:9) , , a ( G , ) b ( s , ,a ) c ( s , ,b ) Fig. 1.
The 0 , Among the strategy profiles, one can select specific ones that are called subgameperfect equilibria . Subgame perfect equilibria are specific strategy profiles thatfulfill a predicate
SPE . This predicate relies on another predicate PE which checksa local property. PE ( s ) ⇔ s ⇓ ∧ s = hh p, d, s d , s r ii ⇒ b s d ( p ) ≥ b s r ( p ) ∧ s = hh p, r, s d , s r ii ⇒ b s r ( p ) ≥ b s d ( p )A strategy profile is a subgame perfect equilibrium if the property PE holdsalways: SPE = (cid:3) PE . We may now wonder what the subgame perfect equilibria of the 0,1-game are.We present two of them in Figure 1.b and 1.c. But there are others. To presentthem, let us define a predicate “ A continues and B eventually stops” AcBes ( s ) k s i , + s = hh p, c, hh f ii , s ′ ii ⇒ ( p = A ∧ f = f , ∧ c = r ∧ AcBes ( s ′ )) ∨ ( p = B ∧ f = f , ∧ ( c = d ∨ AcBes ( s ′ )) Proposition 5. ( S ( s ) ∨ S ( s )) ⇒ AcBes ( s ) ⇒ b s = f , Proof. If s = hh p, c, hh f ii , s ′ ii , then S ( s ′ ) ∨ S ( s ′ ). Therefore if AcBes ( s ′ ), byinduction, b s ′ = f , . By case: – If p = A ∧ c = r , then AcBes ( s ′ ) and by definition of b s , we have b s = b s ′ = f , – if p = B ∧ c = d , the b s = \ hh f , ii = f , . – if p = B ∧ c = r , , then AcBes ( s ′ ) and by definition of b s , b s = b s ′ = f , .Like we generalize PE to SPE by applying the modality (cid:3) , we generalize
AcBes into
SAcBes by stating:
SAcBes = (cid:3) AcBes . here are at least two profiles which satisfies SAcBes namely s , ,a and s , ,b which have been studied in [14] and pictured in Figure 1: s , ,a k s c , + hh A , r, f , , s , ,b ii s , ,a k s c , + hh A , d, f , , s , ,b ii s , ,b k s c , + hh B , d, f , , s , ,a ii s , ,b k s c , + hh B , r, f , , s , ,a ii Proposition 6.
SAcBes ( s ) ⇒ s ⇓ . We may state the following proposition.
Proposition 7. ∀ s, ( S ( s ) ∨ S ( s )) ⇒ ( SAcBes ( s ) ⇒ SPE ( s )) . Proof.
Since
SPE is a coinductively defined predicate, the proof is by coinduction.Given an s , we have to prove ∀ s, (cid:3) AcBes ( s ) ∧ ( S ( s ) ∨ S ( s )) ⇒ (cid:3) PE ( s ) . For that we assume (cid:3)
AcBes ( s ) ∧ ( S ( s ) ∨ S ( s )) and in addition (coinductionprinciple) (cid:3) PE ( s ′ ) for all strict subprofiles s ′ of s and we prove PE ( s ). In otherwords, s ⇓ ∧hh p, d, s d , s r ii ⇒ b s d ( p ) ≥ b s r ( p ) ∧ hh p, r, s d , s r ii ⇒ b s r ( p ) ≥ b s d ( p ) . By Proposition 6, we have s ⇓ .By Proposition 5, we know that for every subprofile s ′ of a profile s thatsatisfies S ( s ) ∨ S ( s ) we have b s ′ = f , except when s ′ = hh f , ii . Let us prove hh p, d, s d , s r ii ⇒ b s d ( p ) ≥ b s r ( p ) ∧ hh p, r, s d , s r ii ⇒ b s r ( p ) ≥ b s d ( p ) . Let us proceedby case: – s = hh A , r, hh f , ii , s ′ ii . Then S ( s ) and S ( s ′ ). Since (cid:3) AcBes ( s ), we have AcBes ( s ′ ), therefore b s ′ = f , hence b s ′ ( A ) = 1 and f , ( A ) = 0, henceforth b s ′ ( A ) ≥ f , ( A ) . – s = hh B , r, hh f , ii , s ′ ii . Then S ( s ) and S ( s ′ ). Since (cid:3) AcBes ( s ), we have AcBes ( s ′ ), therefore b s ′ = f , hence b s ′ ( B ) = 0 and f , ( B ) = 0, henceforth b s ′ ( B ) ≥ f , ( B ) . Symmetrically we can define a predicate
BcAes for “ B continues and A eventuallystops” and a predicate SBcAes which is
SBcAes = (cid:3) BcAes which means that B continues always and A stops infinitely often. With the same argument as for SAcBes one can conclude that ∀ s, ( S ( s ) ∨ S ( s )) ⇒ SBcAes ( s ) ⇒ SPE ( s ) . We claim that
SAcBes ∨ SBcAes fully characterizes
SPE of 0,1-games, in otherwords.
Conjecture 1. ∀ s, ( S ( s ) ∨ S ( s )) ⇒ ( SAcBes ( s ) ∨ SBcAes ⇔ SPE ( s )) . Before talking about escalation, let us see the connection between subgame per-fect equilibrium and Nash equilibrium in a sequential game. In [17], the definitionof a Nash equilibrium is as follows:
A Nash equilibrium is a“pattern[s] of behaviorith the property that if every player knows every other player’s behavior she hasnot reason to change her own behavior” in other words, “a Nash equilibrium [is]a strategy profile from which no player wishes to deviate, given the other player’sstrategies.” . The concept of deviation of agent p is expressed by a binary relationwe call convertibility and we write ⊢ p ⊣ . It is defined inductively as follows: s ∼ s s ′ s ⊢ p ⊣ s ′ s ⊢ p ⊣ s ′ s ⊢ p ⊣ s ′ hh p, c, s , s ii ⊢ p ⊣ hh p, c ′ , s ′ , s ′ ii s ⊢ p ⊣ s ′ s ⊢ p ⊣ s ′ hh p ′ , c, s , s ii ⊢ p ⊣ hh p ′ , c, s ′ , s ′ ii We define the predicate
Nash as follows:
Nash ( s ) ⇔ ∀ p, ∀ s ′ , s ⊢ p ⊣ s ′ ⇒ b s ( p ) ≥ b s ′ ( p ′ ) . The concept of Nash equilibrium is more general than that of subgame perfectequilibrium and we have the following result:
Proposition 8.
SPE ( s ) ⇒ Nash ( s ) . The result has been proven in COQ and we refer to the script (see[15]): http://perso.ens-lyon.fr/pierre.lescanne/COQ/EscRatAI/http://perso.ens-lyon.fr/pierre.lescanne/COQ/EscRatAI/SCRIPTS/
Notice that we defined the convertibility inductively, but a coinductive def-inition is possible. But this would give a more restrictive definition of Nashequilibrium.
Escalation in a game with a set P of agents occurs when there is a tuple of strate-gies ( st p ) p ∈ P such that its sum is not convergent, in other words, ¬ ( M p ∈ P st p ) ↓ .Said differently, it is possible that the agents have all a private strategy whichcombined with those of the others makes a strategy profile which is not con-vergent, which means that the strategy profile goes to infinity when followingthe choices. Notice the two uses of a strategy profile: first, as a subgame perfectequilibrium, second as a combination of the strategies of the agents.Consider the strategy: st A , ∞ = J r, J f , K , st ′ A , ∞ K st ′ A , ∞ = J B , J f , K , st A , ∞ K This should be called perhaps feasibility following [18] and [13] nd its twin st B , ∞ = J A , J f , K , st ′ B , ∞ K st ′ B , ∞ = J r, J f , K , st B , ∞ K . Moreover, consider the strategy profile: s A , ∞ = hh A , r, hh f , ii , s B , ∞ ii s B , ∞ = hh B , r, hh f , ii , s A , ∞ ii . Proposition 9.– st A , ∞ . A , – st B , ∞ . B , – st2g ( st A , ∞ , A ) = st2g ( st B , ∞ , B ) = G , , – game ( s A , ∞ ) = G , , – st A , ∞ ⊕ st B , ∞ = s A , ∞ , – ¬ s A , ∞ ↓ .Proof. By coinduction. st A , ∞ and st B , ∞ are both rational since they are built using choices, namely r ,dictated by subgame perfect equilibria which start with r . Another feature ofthis example is that no agent has a clue for what strategy the other agent isusing. Indeed after k steps, A does not know if B has used a strategy derived ofequilibria in SAcBes or in
SBcAes . In other words, A does know if B will stopeventually or not and vice versa. The agents can draw no conclusion of whatthey observe. If each agent does not believe in the threat of the other she isnaturally led to escalation. Acknowledgements
The author thanks Samson Abramsky, Franck Delaplace, Stephane Leroux, Mat-thieu Perrinel, Ren´e Vestergaard, Viktor Winschel for their help, encouragementsand discussions during this research.
In this paper, we have shown how to use coinduction in a field, namely economics,where it has not been used yet, or perhaps in a really hidden form, which hasto be unearthed. We foresee a future for this tool and a possible way for arefoundation of economics. Our choice of rationality is this of a subgame perfect equilibrium, as it generalizesbackward induction, which is usually accepted as the criterion of rationality for finitegame. eferences
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A Finite 0,1 games and the “cut and extrapolate”method
We spoke about the “cut and extrapolate” method, applied in particular to thedollar auction. Let us see how it would work on the 0,1-game. Finite games, finitestrategy profiles and payoff functions of finite strategy profiles are the inductiveequivalent of infinite games, infinite strategy profiles and infinite payoff functionswhich we presented. Notice that payoff functions of finite strategy profiles arealways defined. Despite we do not speak of the same types of objects, we use thesame notations, but this does not lead to confusion. Consider two finite familiesof finite games, that could be seen as approximations of the 0,1-game: F , = h A , h f , i , h B , h f , i , F , ii ∪ {h f , i} K , = h A , h f , i , K ′ , i K ′ , = h B , h f , i , K , i ∪ {h f , i} In F , we cut after B and replace the tail by h f , i . In K , we cut after A andreplace the tail by h f , i . Recall [23] the predicate BI , which is the finite versionof PE . BI ( h f i ) BI ( h p, c, s d , s r i ) = BI ( s l ) ∧ BI ( s r ) ∧hh p, d, s d , s r ii ⇒ b s d ( p ) ≥ b s r ( p ) ∧hh p, r, s d , s r ii ⇒ b s r ( p ) ≥ b s d ( p )We consider the two families of strategy profiles: SF , ( s ) k s i , + ( s = hh A , d ∨ r, hh f , ii , hh B , r, hh f , ii , s ′ iiii ∧ SF , ( s ′ ) ∨ s = hh f , ii SK , ( s ) k s i , + s = hh A , r, hh f , ii , s ′ ii ∧ SK ′ , ( s ′ ) SK ′ , ( s ) k s i , + ( s = hh B , d, hh f , ii , s ′ ii ∨ s = hh B , r, hh f , ii , s ′ ii ) ∧ SK , ( s ′ ) ∨ s = hh f , ii In SF , , B continues and A does whatever she likes and in SK , , A continues and B does whatever she likes. The following proposition characterizes the backwardinduction equilibria for games in F , and K , respectively and is easily provedby induction: roposition 10.– game ( s ) ∈ F , ∧ SF , ( s ) ⇔ BI ( s ) , – game ( s ) ∈ K , ∧ SK , ( s ) ⇔ BI ( s ) . This shows that cutting at an even or an odd position does not give the samestrategy profile by extrapolation. Consequently the “cut and extrapolate” methoddoes not anticipate all the subgame perfect equilibria. Let us add that whencutting we decide which leaf to insert, namely h f , i or h f , i , but we could doanother way around obtaining different results. To avoid escalation in the dollar auctions, somerequire a limited payrolllimited payroll