Many-player games of optimal consumption and investment under relative performance criteria
MMANY-PLAYER GAMES OF OPTIMAL CONSUMPTION ANDINVESTMENT UNDER RELATIVE PERFORMANCE CRITERIA
DANIEL LACKER AND AGATHE SORET
Abstract.
We study a portfolio optimization problem for competitive agents with CRRAutilities and a common finite time horizon. The utility of an agent depends not only on herabsolute wealth and consumption but also on her relative wealth and consumption whencompared to the averages among the other agents. We derive a closed form solution forthe n -player game and the corresponding mean field game. This solution is unique in theclass of equilibria with constant investment and continuous time-dependent consumption,both independent of the wealth of the agent. Compared to the classical Merton problemwith one agent, the competitive model exhibits a wide range of highly nonlinear andnon-monotone dependence on the agents’ risk tolerance and competitiveness parameters.Counter-intuitively, competitive agents with high risk tolerance may behave like non-competitive agents with low risk tolerance. Introduction
In this paper, we extend the CRRA model for the optimal investment problem recentlydeveloped by Lacker and Zariphopoulou [21] to include consumption. Our model can beloosely described as follows, with full details given in Section 2. Each agent chooses con-sumption and investment policies, with access to a riskless bond and a lognormal stock.The stocks in which the different agents specialize can be correlated, and we cover theextreme case of perfect correlation, i.e., a single stock in which all agents trade. Eachagent has a CRRA utility depending on both absolute and relative wealth at the (common)time horizon T as well as absolute and relative consumption, the latter in a time-integratedsense. Agents have different levels of risk aversion and different preferences toward absoluteversus relative performance. The relative performance criteria couple these n optimizationproblems, and we find the (unique, in a sense to be clarified later) Nash equilibrium interms of the various model parameters. As is natural in light of the classical Merton prob-lem [23], in equilibrium each agent invests a constant fraction of wealth in the stock, andthe consumption strategy is time-dependent but independent of the agent’s wealth.The equilibrium behavior fits with Samuelson’s result [26]: the investment strategy isindependent of the consumption strategy. That is, the investment strategy is exactly thesame as in the model without consumption studied in [21]. The equilibrium consumptionpolicy, as a function of the various model parameters, displays even more highly nonlinearand non-monotone behavior than the investment policy, and we study this in detail inSection 4. Notably, each agent’s rate of consumption c t changes monotonically with time t over the entire horizon [0 , T ]; however, whether an agent increases or decreases consumptionover time depends in a complex manner on her own risk preferences as well as certainaggregates of the other agents’ parameters.Three key features of our model are relative consumption concerns, relative wealthconcerns, and asset specialization. We defer to the introduction of [21] for a thoroughdiscussion of the latter two topics and further references, but we stress the particularlyimportant and by now well-established point that mutual fund choice is highly influencedby relative performance [28]. That is, out-performing other fund managers tends to attract a r X i v : . [ q -f i n . M F ] M a y DANIEL LACKER AND AGATHE SORET greater future investment in one’s own fund. The most closely related works to ours,after [21], are [2, 3, 5, 11, 12, 14]. These papers study continuous-time models of optimalinvestment under relative performance concerns in various settings, including different kindsof utilities, equilibrium pricing, and state constraints, but none incorporate consumption.There are two natural arguments for studying relative consumption concerns. On theone hand, interpreting agents as fund managers, we may think of consumption as capitalaccumulation, in the form of equipment, technology, or benefits for employees. A highrelative consumption in this sense would naturally attract clientele or better employees,and more generally it should lead to similar benefits as a high relative wealth. On theother hand, if we interpret the agents in our model as household investors, then relativeconsumption concerns fit naturally with models of keeping up with the Joneses ; this line ofliterature directly incorporates the social aspects of investment and consumption decisions[1, 9, 10, 15].Our paper contributes to the literatures on optimal consumption and investment as wellas the application of mean field games. The dynamic problem of lifetime consumption andinvestment planning with one player was formalized and studied in the landmark papersof Merton [23, 24] and Samuelson [26]. Later work incorporated more complex featuresinto the models, such as general price processes, bankruptcy, etc. [19, 20]. Those thatincorporated multiple agents into the model, such as [18, 27], did so in an equilibriumcontext; each agent’s behavior depends on the others only through the price, which isdetermined in equilibrium. Agents are price-takers in our model, and we do not attemptto incorporate price equilibrium, as this would severely strain tractability.In another direction, our work provides a new explicitly solvable mean field game model.Mean field games, introduced in [17, 22], are rarely explicitly solvable outside of linear-quadratic examples. See [4, 6, 16, 21, 29] for some notable exceptions and the book [7]for further background on the active area of mean field games. From a mean field gameperspective, our model is rather complex: It involves common noise, degenerate volatilitycoefficients, singular objective functions, and a mean field interaction through both thestates and controls (i.e., an extended mean field game [7, Chapter I.4.6]). Nevertheless,the precise structure of the problem lends itself to an explicit solution. Our argumentfollows along the lines of [21], treating the mean field term (geometric mean of wealth)as a state variable, which leads to a fixed point problem involving a single Hamilton-Jacobi-Bellman (HJB) equation as opposed to the n -dimensional HJB system often usedfor stochastic differential games. After showing that this equation admits a unique andseparable classical solution, the fixed point is resolved via a system of non-linear ordinarydifferential equations. Despite many similarities with [21], the consumption renders thearguments substantially more involved.The paper is organized as follows. In Section 2, we formulate and solve the n -agentmodel described above. Then, in Section 3, we study the infinite population counterpartof this problem, arguing that the n → ∞ limit results in a simpler form of the equilibrium.Finally, Section 4 discusses and interprets the form of the equilibrium and its dependenceon the model parameters. 2. The n-agent game
In this section, we consider the n -player game, where each agent trades in a commoninvestment horizon [0 , T ]. Agents may invest in their own specific stocks or in a commonriskless bond which offers zero interest rate. The price of stock i , in which only agent i ONSUMPTION AND INVESTMENT UNDER RELATIVE PERFORMANCE CRITERIA 3 trades, is given by the dynamics dS it S it = µ i dt + ν i dW it + σ i dB t , (1)where the Brownian motions W , ..., W n , B are independent and defined on a probabilityspace (Ω , F , P ), which we endow with the natural filtration ( F t ) t ∈ [0 ,T ] generated by these n + 1 Brownian motions, and where the market parameters are constants µ i > σ i ≥ ν i ≥
0, with σ i + ν i >
0. The prices S it are assumed to be one-dimensional for simplicity,although we could easily extend the results to k -dimensional prices.This setup covers the important special case of a single stock , corresponding to thesituation where all the stocks are identical. That is, µ i = µ , ν i = 0, and σ i = σ , for all i = 1 , ..., n and for some µ, σ > i , and so S i ≡ S j for each i, j (assuming theinitial values agree). In the single stock case, the agents face the same market opportunitiesrather than specializing in different assets, though their risk preferences still differ.Each agent i chooses a self-financing strategy, ( π it ) t ∈ [0 ,T ] , denoting the proportion ofwealth invested in the stock i , and a consumption policy, ( c it ) t ∈ [0 ,T ] . The wealth process ofagent i is then given by dX it = π it X it (cid:0) µ i dt + ν i dW it + σ i dB t (cid:1) − c it X it dt, X i = x i . (2)Note that c it X it represents the instantaneous rate of consumption of agent i , so that c it is the rate per unit wealth. We say that a portfolio strategy is admissible if it belongsto the set A of F -progressively measurable R × R + -valued process ( π t , c t ) t ∈ [0 ,T ] satisfying E (cid:82) T ( π t + c t ) dt < ∞ . Throughout the paper, R + := (0 , ∞ ) denotes the strictly positivereals, and we do not allow a consumption rate of zero. This is reasonable and less restrictivethan it may at first appear because the form of our utility functions, introduced in thenext paragraph, will ensure that agents’ marginal utilities approach + ∞ as consumptionapproaches zero. Note also that for any admissible portfolio strategy we have X it > t ∈ [0 , T ]. Indeed, we parametrized our consumption process as we did in (2) in part toavoid the possibility of bankruptcy and in part to avoid imposing any state constraints.The utility function of each agent belongs to the family of power (CRRA) utilities, U ( x ; δ ) = (cid:40) − /δ x − /δ if δ (cid:54) = 1log x if δ = 1 , defined for x, δ >
0. Agent i seeks to maximize the expected utility J i (( π i , c i ) ni =1 ) = E (cid:20)(cid:90) T U (cid:16) c it X it ( cX t ) − θ i ; δ i (cid:17) dt + (cid:15) i U (cid:16) X iT X − θ i T ; δ i (cid:17)(cid:21) , (3)defined for any vector of admissible strategies ( π i , c i ) ni =1 where ( π i , c i ) ∈ A for each i =1 , . . . , n . Here X T = (cid:0)(cid:81) nk =1 X kT (cid:1) /n and cX t = (cid:0)(cid:81) nk =1 ( c t X t ) k (cid:1) /n are the population(geometric) average wealth and consumption rate, respectively. The parameters δ i > θ i ∈ [0 ,
1] represent respectively the i th agent’s risk tolerance and competition weight. Weapply the same utility function to both wealth and consumption for tractability reasons, butwe scale the utility of wealth with the parameter (cid:15) i > DANIEL LACKER AND AGATHE SORET the utility function, revealed by writing the terms inside the utility function as c it X it ( cX t ) − θ i = ( c it X it ) − θ i (cid:18) c it X it cX t (cid:19) θ i , X iT X − θ i T = ( X iT ) − θ i (cid:18) X iT X T (cid:19) θ i . The ratios c it X it /cX t and X iT /X T measure the relative consumption rate and relative termi-nal wealth, respectively. In particular, the utility function in (3) is applied to the log-convexcombination between absolute and relative consumption rate and terminal wealth, with θ i controlling the tradeoff between absolute and relative performance. For θ i close to 1, agent i is more concerned with relative performance than absolute performance, and for θ i = 0agent i is not at all competitive and ignores the rest of the population.The goal is to find a Nash equilibrium, an investment strategy ( (cid:126)π ∗ t , (cid:126)c ∗ t ) t ∈ [0 ,T ] such that π i, ∗ t and c i, ∗ t are respectively the optimal stock and consumption allocation exercised byagent i in response to the strategy of all the other agents. With Merton’s problem and therecent findings of [21] in mind, we might expect to find an equilibrium where the investmentstrategies (cid:126)π ∗ t are constant and the consumption strategies c i, ∗ t are only time-dependent. Definition 2.1.
We say that a vector ( π i, ∗ , c i, ∗ ) ni =1 of admissible strategies (i.e., ( π i, ∗ , c i, ∗ ) ∈A for each i ) is an equilibrium if for each i = 1 , ..., n and each ( π, c ) ∈ A we have J i (( π i, ∗ , c i, ∗ ) ni =1 ) ≥ J i (cid:16) . . . , ( π i − , ∗ , c i − , ∗ ) , ( π, c ) , ( π i +1 , ∗ , c i +1 , ∗ ) , . . . (cid:17) . An equilibrium ( π i, ∗ , c i, ∗ ) ni =1 is called a strong equilibrium if, for each i , the process c i, ∗ isdeterministic and continuous, and the process π i, ∗ is deterministic and constant. The main result is the following, which gives the explicit form of an equilibrium:
Theorem 2.2.
Let n ≥ . Assume that for all i = 1 , ..., n , we have x i > , δ i > , θ i ∈ [0 , T ] , (cid:15) i > , µ i > , σ i ≥ , ν i ≥ , and σ i + ν i > . Then there is a unique strongequilibrium ( π i, ∗ , c i, ∗ ) ni =1 , and it takes the following form: π i, ∗ = δ i µ i σ i + ν i (1 + ( δ i − θ i /n ) − θ i ( δ i − σ i σ i + ν i (1 + ( δ i − θ i /n ) φ ψ (4) c i, ∗ t = (cid:16) β i + (cid:16) λ i − β i (cid:17) e − β i ( T − t ) (cid:17) − if β i (cid:54) = 0( T − t + λ − i ) − if β i = 0 . (5) The constants ( φ, ψ ) and ( β i , λ i ) ni =1 are given by φ = 1 n n (cid:88) k =1 δ k µ k σ k σ k + ν k (1 + ( δ k − θ k /n ) ,ψ = 1 n n (cid:88) k =1 θ k ( δ k − σ k σ k + ν k (1 + ( δ k − θ k /n ) ,β i = θ i ( δ i − n (cid:80) nk =1 δ k ρ k n (cid:80) nk =1 θ k ( δ k − − δ i ρ i ,λ i = (cid:15) − δ i i (cid:32) n (cid:89) k =1 (cid:15) δ k k (cid:33) /n θ i ( δ i − / (1+ n (cid:80) nk =1 θ k ( δ k − , (6) This definition of equilibrium is more specifically of open-loop type, but a strong equilibrium, beingnonrandom, can be shown to also provide an equilibrium over closed-loop or Markovian controls.
ONSUMPTION AND INVESTMENT UNDER RELATIVE PERFORMANCE CRITERIA 5 where we define also ( ρ i ) ni =1 by ρ i = (1 − /δ i ) (cid:40) (1 − θ i /n )( µ i − σ i θ i (1 − /δ i ) n (cid:80) k (cid:54) = i σ k π k, ∗ ) σ i + ν i )(1 − (1 − θ i /n )(1 − /δ i ))+ 12 (cid:16)(cid:16) n (cid:88) k (cid:54) = i σ k π k, ∗ (cid:17) + 1 n (cid:88) k (cid:54) = i ( ν k π k, ∗ ) (cid:17) θ i (1 − /δ i ) − θ i n (cid:88) k (cid:54) = i µ k π k, ∗ + θ i n (cid:88) k (cid:54) = i ( σ k + ν k )( π k, ∗ ) (cid:41) , Moreover, we have the identity n n (cid:88) k =1 σ k π k, ∗ = φ ψ . (7)Note that δ i = 1 implies β i = 0, which means that log-investors always use the secondform of c i, ∗ t given in (5). The form of the equilibrium does not seem to simplify muchfurther, except in the single stock case: Corollary 2.3. (Single stock) Assume that for all i = 1 , ..., n we have µ i = µ > , σ i = σ > , and ν i = 0 . Then there is a unique strong equilibrium ( π i, ∗ , c i, ∗ ) ni =1 , and it takes thefollowing form: π i, ∗ = µσ (cid:18) δ i − θ i θ crit ( δ i − (cid:19) (8) c i, ∗ t = (cid:16) β i + (cid:16) λ i − β i (cid:17) e − β i ( T − t ) (cid:17) − if β i (cid:54) = 0 , ( T − t + λ − i ) − if β i = 0 . (9) For each i , the constant λ i is given by (6) , and β i and θ crit are given by β i = µ σ (1 − δ i ) (cid:18) − θ i θ crit (cid:19) (cid:18) δ i − θ i θ crit ( δ i − (cid:19) ,θ crit = 1 + n (cid:80) nk =1 θ k ( δ k − n (cid:80) nk =1 δ k . In Section 3 we simplify this further by sending n → ∞ . Then, in Section 4, we analyzein detail how the equilibrium behavior depends on the various parameters. Proof of Theorem 2.2.
First, we fix an agent i ∈ { , . . . , n } and suppose that all otherplayers follow given strategies. That is, for k (cid:54) = i , let π k ∈ R and c k : [0 , T ] → R + denotefixed admissible strategies for the other agents, in which the investment policy π k is constantand the consumption policy c k is a deterministic continuous function. We will solve theoptimization problem for agent i , determining the agent’s best response to the competitors’strategies. Then, we will resolved the resulting fixed point problem.Define Y t := ( (cid:81) k (cid:54) = i X kt ) /n , where X kt solves (2) subject to the strategies ( π k , c k ), with X k = x k . We use the following abbreviations:Σ k = σ k + ν k (cid:99) µπ − i = n (cid:80) k (cid:54) = i µ k π k , (cid:99) σπ − i = n (cid:80) k (cid:54) = i σ k π k , (cid:100) Σ π − i = n (cid:80) k (cid:54) = i Σ k π k , (cid:92) ( νπ ) − i = n (cid:80) k (cid:54) = i ν k π k , (cid:98) c − i ( t ) = n (cid:80) k (cid:54) = i c k ( t ) . DANIEL LACKER AND AGATHE SORET
A straightforward calculation with Itˆo’s formula (cf. the proof of Theorem 14 in [21]) showsthat the process Y t satisfies dY t Y t = ( η i − (cid:98) c − i ( t )) dt + 1 n (cid:88) k (cid:54) = i ν k π k dW kt + (cid:99) σπ − i dB t , (10)where we define also η i = (cid:99) µπ − i − (cid:18) (cid:100) Σ π − i − (cid:99) σπ − i − n (cid:92) ( νπ ) − i (cid:19) . The i th agent then solves the optimization problemsup ( π i ,c i ) ∈A E (cid:20)(cid:90) T U (cid:16) ( c it X it ) − θ i /n (¯ c − i ( t ) Y t ) − θ i ; δ i (cid:17) dt + (cid:15) i U (cid:16)(cid:0) X iT (cid:1) − θ i /n Y − θ i T ; δ i (cid:17)(cid:21) , (11)where ¯ c − i ( t ) = (cid:16)(cid:81) k (cid:54) = i c k ( t ) (cid:17) /n and dX it = π it X it ( µ i dt + ν i dW it + σ i dB t ) − c it X it dt, X i = x i , with ( Y t ) t ∈ [0 ,T ] solving (10). Treating ( X i , Y ) as the state process, we solve this stochasticoptimal control problem by noting that the value (11) should equal v ( X i , Y , v ( x, y, t ) solves the HJB equation0 = v t + sup π ∈ R (cid:20) π ( µ i xv x + σ i (cid:99) σπ − i xyv xy ) + 12 π Σ i x v xx (cid:21) + sup c ∈ R + (cid:104) − cxv x + U (cid:16) ( cx ) (1 − θ i /n ) (¯ c − i ( t ) y ) − θ i ; δ i (cid:17)(cid:105) + ( η i − (cid:98) c − i ( t )) yv y + 12 (cid:18) n (cid:92) ( νπ ) − i + (cid:99) σπ − i (cid:19) y v yy , (12)for ( x, y, t ) ∈ R + × R + × [0 , T ), with terminal condition v ( x, y, T ) = (cid:15) i U ( x − θ i /n y − θ i ; δ i ) . (13)Notice that the two suprema in (12) are finite if v xx < v x >
0, so we assume for themoment that this is the case, and we will ultimately check that our solution does satisfythese constraints. If follows from a standard verification theorem that there can be at mostone classical solution of this PDE. Since the utility takes a different form depending onwhether or not δ i = 1, we treat these two cases separately in the next part of the proof. The case δ i (cid:54) = 1 : The utility function takes the form U (cid:16) ( cx ) (1 − θ i /n ) (¯ c − i ( t ) y ) − θ i ; δ i (cid:17) = (cid:18) − δ i (cid:19) − ( cx ) (1 − θ i /n )(1 − /δ i ) (¯ c − i ( t ) y ) − θ i (1 − /δ i ) . Applying the first order conditions, the suprema in (12) are attained by π i, ∗ ( x, y, t ) = − µ i xv x ( x, y, t ) + σ i (cid:99) σπ − i xyv xy ( x, y, t )Σ i x v xx ( x, y, t ) , (14)and c i, ∗ ( x, y, t ) = 1 x (cid:32) (1 − θ i n )(¯ c − i ( t ) y ) − θ i (1 − /δ i ) v x ( x, y, t ) (cid:33) − (1 − θi/n )(1 − /δi ) . (14’) We use a bar c to denote a geometric average and a hat (cid:98) c to denote an arithmetic average. ONSUMPTION AND INVESTMENT UNDER RELATIVE PERFORMANCE CRITERIA 7
Let us introduce the following constants: γ i = 11 − (1 − θ i /n )(1 − /δ i ) , (15)Γ i = (cid:18) − θ i n (cid:19) γ i (cid:18) γ i − (cid:19) . Using these constants and the expressions (14) and (14’), the HJB equation (12) becomes0 = v t −
12 ( µ i xv x + σ i (cid:99) σπ − i xyv xy ) Σ i x v xx + 12 (cid:0) (cid:99) σπ − i + 1 n (cid:92) ( νπ ) − i (cid:1) y v yy + ( η i − (cid:98) c − i ( t )) yv y + ( v x ) − γ i (¯ c − i ( t ) y ) − γ i θ i (1 − /δ i ) Γ i . (16)We now make the ansatz v ( x, y, t ) = (cid:15) i (cid:18) − δ i (cid:19) − x (1 − θ i /n )(1 − /δ i ) y − θ i (1 − /δ i ) f i ( t ) , (17)for a differentiable function f i : [0 , T ] → R to be determined. Note that the boundarycondition (13) requires f i ( T ) = 1. Plugging this into the HJB equation (16), we find that v ( x, y, t ) /f i ( t ) factors out of each term, and we get0 = f (cid:48) i ( t ) + (cid:18) ρ i + θ i (cid:18) − δ i (cid:19) (cid:98) c − i ( t ) (cid:19) f i ( t ) + (cid:15) − γ i i γ i ¯ c − i ( t ) − γ i θ i (1 − /δ i ) f i ( t ) − γ i , (18)where we define ρ i = (cid:18) − δ i (cid:19) (cid:18) γ i (1 − θ i /n )( µ i − σ i (cid:99) σπ − i θ i (1 − /δ i )) i + 12 ( (cid:99) σπ − i + 1 n (cid:92) ( νπ ) − i ) θ i (1 − /δ i ) − θ i (cid:99) µπ − i + θ i (cid:100) Σ π − i (cid:19) . (19)Indeed, the last term in (18) comes from the identity v x ( x, y, t ) − γ i y − γ i θ i (1 − /δ i ) = (cid:15) − γ i i (cid:18) − θ i n (cid:19) − γ i (cid:18) − δ i (cid:19) f i ( t ) − γ i v ( x, y, t ) . To solve (18), let us for the moment abbreviate a i ( t ) := ρ i + θ i (1 − /δ i ) (cid:98) c − i ( t ) , b i ( t ) := (cid:15) − γ i i γ i ¯ c − i ( t ) − γ i θ i (1 − /δ i ) . (20)Then (18) rewrites as f (cid:48) i + a i f i + b i f − γ i i = 0 . This is an example of a
Bernoulli equation , and a well known change of variables leads tothe solution. Indeed, and divide by f − γ i i (after noting that γ i >
0) and use the substitution u i ( t ) = f γ i i ( t ), so that (18) becomes the linear differential equation1 γ i u (cid:48) i + a i u i + b i = 0 , with terminal condition u i ( T ) = 1. This linear equation admits the unique solution u i ( t ) = e γ i (cid:82) Tt a i ( s ) ds + (cid:90) Tt γ i b i ( s ) e − γ i (cid:82) st a i ( r ) dr ds. DANIEL LACKER AND AGATHE SORET
Note that b i is positive everywhere, and thus so is u i . Hence, u /γ i i is well defined, and theunique solution to (18) is given by f i ( t ) = (cid:18) e γ i (cid:82) Tt a i ( s ) ds + (cid:90) Tt γ i b i ( s ) e − γ i (cid:82) st a i ( r ) dr ds (cid:19) /γ i . (21)Substituting this solution (21) into the ansatz (17) yields the solution v ( x, y, t ) of the HJBequation, as long as we check that v xx < v x >
0. But this is straightforward: v x ( x, y, t ) = (cid:15) i (1 − θ i /n ) x − /γ i y − θ i (1 − /δ i ) f i ( t ) > ,v xx ( x, y, t ) = − (cid:15) i γ i (1 − θ i /n ) x − /γ i − y − θ i (1 − /δ i ) f i ( t ) < , where we again use γ i >
0. Therefore, in terms of f i , we may express the optimal controlsfrom (14) and (14’) as π i, ∗ = γ i ( µ i − σ i (cid:99) σπ − i θ i (1 − /δ i ))Σ i ,c i, ∗ t = (cid:15) − γ i i (¯ c − i ( t )) − γ i θ i (1 − /δ i ) f − γ i i ( t ) . (22) The case δ i = 1 : In the case δ i = 1 we must proceed differently, but we will ultimatelyderive optimal controls that are consistent with the formulas in (22). Note first that wemay greatly simplify the form of (11), because the logarithmic utility function implies inparticular that the other players no longer influence player i ’s optimization. That is, player i maximizes the simplified objective(1 − θ i /n ) E (cid:20)(cid:90) T log( c it X it ) dt + (cid:15) i log X iT (cid:21) . (23)Noting that 1 − θ i /n >
0, the value (23) is equal to (1 − θ i /n ) w ( X i , w ( x, t ) solvesthe HJB equation0 = w t + max π ∈ R (cid:20) πµ i xw x + π
12 Σ i x w xx (cid:21) + max c ∈ R + [ − cxw x + log( cx )] , (24)for ( x, t ) ∈ R + × [0 , T ), with terminal condition w ( x, T ) = (cid:15) i log x . The maximum isattained by π i, ∗ ( x, t ) = − µ i xw x ( x, t )Σ i x w xx ( x, t ) , c i, ∗ ( x, t ) = 1 xw x ( x, t ) . (25)The HJB equation (24) then becomes0 = w t −
12 ( µ i xw x ) Σ i x w xx − − log w x . (26)Make the ansatz w ( x, t ) = f i ( t ) (cid:15) i log x + g i ( t ) , where f i and g i are to be determined and satisfy f i ( T ) = 1 and g i ( T ) = 0. Plug this into(26), defining the constant (cid:98) ρ i = µ i (cid:15) i / i , to get (cid:0) (cid:15) i f (cid:48) i ( t ) + 1 (cid:1) log x + g (cid:48) i ( t ) + (cid:98) ρ i f i ( t ) − − log (cid:15) i − log f i ( t ) = 0 . Since there is only one term depending on x , we must have (cid:15) i f (cid:48) i ( t ) + 1 = 0 , f i ( T ) = 1 . This yields f i ( t ) = (cid:15) − i ( T − t ) + 1. Then g i must solve g (cid:48) i ( t ) = − (cid:98) ρ i ( (cid:15) − i ( T − t ) + 1) + 1 + log(( T − t ) + (cid:15) i ) , ONSUMPTION AND INVESTMENT UNDER RELATIVE PERFORMANCE CRITERIA 9 which is easily integrated using g i ( T ) = 0 to get the solution g i , though we will not need touse it explicitly. Note also that f i >
0, and thus w x > w xx < δ i = 1. Recalling (25), we deduce that the optimal controlsare π i, ∗ = µ i Σ i , c i, ∗ t = 1 T − t + (cid:15) i . (27)Note that the result (22) obtained above specializes to (27) when δ i = 1. Indeed, if δ i = 1,then ρ i = 0 and γ i = 1, and the functions a i and b i defined in (20) reduce to a i ≡ b i ≡ /(cid:15) i . Thus (21) becomes f i ( t ) = (cid:15) − i ( T − t ) + 1, and (22) becomes (27). Completing the proof:
We now complete the proof, using the form we found above forthe optimal control of player i in response to the other players’ choices. Namely, the bestresponse of player i is given by the controls in (22), where f i is defined as in (21), and wehave seen that these formulas are valid for both cases δ i = 1 and δ i (cid:54) = 1.Note that if we assume that the other consumption functions are positive and con-tinuous, then the optimal feedback consumption that we have found is also positive andcontinuous on [0 , T ]. Now, to conclude the proof, note that the original choice of ( π i , c i ) ni =1 is a strong equilibrium if and only if for each i = 1 , ..., n , we have π i, ∗ = π i and c i, ∗ t = c i ( t ) ∀ t ∈ [0 , T ] , where ( π i, ∗ , c i, ∗ ) were given in (22).We first address the investment policy. Note that we obtained exactly the same optimalcontrol π i, ∗ as in the problem without consumption, and we can conclude as in the proof of[21, Theorem 14] that π i, ∗ = π i for all i = 1 , ..., n if and only if π i, ∗ is as in (4). In addition,we prove the identity (7) just as in [21, Theorem 14]. Recall that ρ i defined in (19) dependson the investment policies π k of the other agents but not on the consumption policies; inparticular, in equilibrium we have ρ i := (cid:16) − δ i (cid:17)(cid:40) (1 − θ i /n )( µ i − σ i θ i (1 − /δ i ) n (cid:80) k (cid:54) = i σ k π k, ∗ ) σ i + ν i )(1 − (1 − θ i /n )(1 − /δ i ))+ 12 θ i (cid:16) − δ i (cid:17)(cid:32)(cid:16) n (cid:88) k (cid:54) = i σ k π k, ∗ (cid:17) + 1 n (cid:88) k (cid:54) = i ( ν k π k, ∗ ) (cid:33) − θ i n (cid:88) k (cid:54) = i µ k π k, ∗ + θ i n (cid:88) k (cid:54) = i ( σ k + ν k )( π k, ∗ ) (cid:41) (28)Next, we address the consumption policies. In light of our arguments above, in orderto have an equilibrium, we must simultaneously solve the following system of equations, for i = 1 , . . . , n : c i ( t ) = (cid:15) − γ i i (¯ c − i ( t )) − γ i θ i (1 − /δ i ) ( f i ( t )) − γ i (29)0 = f (cid:48) i ( t ) + ( ρ i + θ i (1 − /δ i ) (cid:98) c − i ( t )) f i ( t ) + (cid:15) − γ i i γ i ¯ c − i ( t ) − γ i θ i (1 − /δ i ) f i ( t ) − γ i , (30)with f i ( T ) = 1. Indeed, the first equation gives the best response of agent i in terms ofthe other agents’ strategies (computed in (22)) and the function f i . The second equation isexactly the differential equation which determined f i , which we solved explicitly in termsof the other agents’ strategies in (21). However, now that we have verified the validity ofthe ansatz for v i ( x, y, t ), to resolve the equilibrium it is more convenient to abandon the explicit form for f i and instead solve the equations (29) and (30) simultaneously. To dothis, first plug (29) into the last term of (30) to find f (cid:48) i ( t ) + (cid:18) ρ i + θ i (cid:18) − δ i (cid:19) (cid:98) c − i ( t ) + 1 γ i c i ( t ) (cid:19) f i ( t ) = 0 . Defining the full average (cid:98) c ( t ) = n (cid:80) nk =1 c k ( t ), note that (cid:98) c − i ( t ) = (cid:98) c ( t ) − c i ( t ) /n . Recallingthe definition of γ i in (15), we deduce that f (cid:48) i ( t ) + (cid:18) ρ i + θ i (cid:18) − δ i (cid:19) (cid:98) c ( t ) + 1 δ i c i ( t ) (cid:19) f i ( t ) = 0 . Hence, with f i ( T ) = 1 this leads to f i ( t ) = exp (cid:18)(cid:90) Tt (cid:18) ρ i + θ i (cid:16) − δ i (cid:17)(cid:98) c ( s ) + 1 δ i c i ( s ) (cid:19) ds (cid:19) . (31)Now notice that (29) is equivalent to c i ( t ) − γ i ( θ i /n )(1 − /δ i ) = (cid:15) − γ i i ¯ c ( t ) − γ i θ i (1 − /δ i ) f i ( t ) − γ i , where ¯ c ( t ) denotes the full geometric average, ¯ c ( t ) := ( (cid:81) nk =1 c k ( t )) /n . Hence, recalling thedefinition of γ i in (15), c i ( t ) = ( (cid:15) i f i ( t )) − γi − γi ( θi/n )(1 − /δi ) ¯ c ( t ) − γiθi (1 − /δi )1 − γi ( θi/n )(1 − /δi ) = ( (cid:15) i f i ( t )) − δ i ¯ c ( t ) − θ i ( δ i − . Next, plug in the expression for f i from (31) to get c i ( t ) = (cid:15) − δ i i ¯ c ( t ) − θ i ( δ i − exp (cid:18) − δ i (cid:90) Tt (cid:18) ρ i + θ i (cid:16) − δ i (cid:17)(cid:98) c ( s ) + 1 δ i c i ( s ) (cid:19) ds (cid:19) , which is equivalent to c i ( t ) exp (cid:18)(cid:90) Tt c i ( s ) ds (cid:19) = (cid:15) − δ i i ¯ c ( t ) − θ i ( δ i − e − δ i ρ i ( T − t ) exp (cid:18) − θ i ( δ i − (cid:90) Tt (cid:98) c ( s ) ds (cid:19) . (32)Taking the geometric mean over i = 1 , . . . , n , we get¯ c ( t ) exp (cid:18)(cid:90) Tt (cid:98) c ( s ) ds (cid:19) = (cid:16) (cid:15) δ (cid:17) − ¯ c ( t ) − (cid:92) θ ( δ − e − (cid:98) δρ ( T − t ) exp (cid:18) − (cid:92) θ ( δ − (cid:90) Tt (cid:98) c ( s ) ds (cid:19) , where we defined (cid:92) θ ( δ −
1) := n (cid:80) nk =1 θ k ( δ k − , (cid:98) δρ := n (cid:80) nk =1 δ k ρ k , and (cid:15) δ := (cid:16)(cid:81) nk =1 (cid:15) δ k k (cid:17) /n . Thus, ¯ c ( t ) exp (cid:18)(cid:90) Tt (cid:98) c ( s ) ds (cid:19) = (cid:16) (cid:15) δ (cid:17) − (cid:92) θ ( δ − e − (cid:99) δρ (cid:92) θ ( δ − ( T − t ) . Plugging this expression into (32), we get c i ( t ) exp (cid:18)(cid:90) Tt c i ( s ) ds (cid:19) = λ i e β i ( T − t ) , (33)where we define β i := θ i ( δ i − (cid:98) δρ (cid:92) θ ( δ − − δ i ρ i ,λ i := (cid:15) − δ i i (cid:16) (cid:15) δ (cid:17) θi ( δi − (cid:92) θ ( δ − > . ONSUMPTION AND INVESTMENT UNDER RELATIVE PERFORMANCE CRITERIA 11
Integrate (33) from t to T and take the logarithm to get (cid:90) Tt c i ( s ) ds = (cid:40) log (cid:16) λ i β i (cid:0) e β i ( T − t ) − (cid:1)(cid:17) if β i (cid:54) = 0log ( λ i ( T − t ) + 1) if β i = 0 . (34)This is indeed well defined because, when β i (cid:54) = 0, the function t (cid:55)→ λ i β i (cid:0) e β i ( T − t ) − (cid:1) isdecreasing on [0 , T ] and equal to 1 at t = T . Differentiating both sides, we finally obtain c i ( t ) = (cid:16) β i + (cid:16) λ i − β i (cid:17) e − β i ( T − t ) (cid:17) − if β i (cid:54) = 0( T − t + λ − i ) − if β i = 0 . In summary, we have found the unique solution of the system of equations given in (29)and (30), justifying our ansatz for the HJB equation (12). With a classical solution ofthe HJB equation in hand, by a standard verification argument [13, 25] we conclude thatthe portfolio and consumption policies identified above do indeed provide the unique bestresponses and thus the unique strong equilibrium. (cid:3) The mean field game
We study in this section the limit as n → ∞ of the n -player game analyzed previously,and we explain how the limit can be viewed as the equilibrium outcome of a (mean field)game with a continuum of agents. For each agent i , define the type vector ζ i := ( x i , δ i , θ i , (cid:15) i , µ i , ν i , σ i ) . We now allow these parameters to depend also on n , though we will not burden the nota-tion with an additional index. These type vectors induce an empirical measure, the typedistribution , which is the probability measure on the type space Z := (0 , ∞ ) × (0 , ∞ ) × [0 , × (0 , ∞ ) × (0 , ∞ ) × [0 , ∞ ) × [0 , ∞ ) , given by m n = n (cid:80) nk =1 δ ζ k . Now note that for each agent i , the equilibrium strategy forthe consumption as well as the investment only depends on the agent’s own type vectorand on the distribution m n of the type vectors. Hence, if we assume m n converges weaklyto some limiting probability measure, then we expect the equilibrium outcome to convergein a certain sense.In order to pass to the limit, let us now denote by ( x , δ, θ, (cid:15), µ, ν, σ ) a Z -valued randomvariable, with ν + σ > n → ∞ limiting forms of theconstants φ and ψ defined in Theorem 2.2 are as follows: φ = E (cid:20) δµσσ + ν (cid:21) , ψ = E (cid:20) θ ( δ − σ σ + ν (cid:21) . (35)To identify limiting forms of the remaining quantities in Theorem 2.2, we additionallyremove the i subscript, letting the randomness of the type vector play the role of the namesof the agents. This gives β = θ ( δ − E [ δρ ]1 + E [ θ ( δ − − δρ, (36) and ρ = (cid:18) − δ (cid:19) (cid:40) δ σ + ν ) (cid:18) µ − σ φ ψ θ (1 − /δ ) (cid:19) + 12 (cid:18) φ ψ (cid:19) θ (1 − /δ ) − θ φ ψ E (cid:20) δµ − θ ( δ − σµσ + ν (cid:21) + θ E (cid:34) ( δµ − θ ( δ − σ φ ψ ) σ + ν (cid:35)(cid:41) . (37)The limiting form of λ i is given by λ = (cid:15) − δ (cid:16) e E [ log( (cid:15) − δ ) ] (cid:17) − θ ( δ − E [ θ ( δ − . (38)Indeed, this is determined by noting that (cid:32) n (cid:89) k =1 (cid:15) δ k k (cid:33) /n = exp (cid:32) n n (cid:88) k =1 log( (cid:15) δ k k ) (cid:33) . The equilibrium investment policy of Theorem 2.2 of the representative agent then becomes π ∗ = δµσ + ν − θ ( δ − σσ + ν φ ψ (39)and the consumption policy becomes c ∗ t = (cid:16) β + (cid:16) λ − β (cid:17) e − β ( T − t ) (cid:17) − if β (cid:54) = 0( T − t + λ − ) − if β = 0 . (40)We next illustrate how this strategy arises as the equilibrium of a mean field game. Let(Ω , F , F = ( F t ) t ∈ [0 ,T ] , P ) be a filtered probability space supporting independent Brownianmotions B and W as well as a random type vector ζ = ( ξ, δ, θ, (cid:15), µ, ν, σ ) as above. Assumethat F is the minimal complete filtration with respect to which ζ is F -measurable and W and B are F -Brownian motions. The representative agent’s wealth process is determinedby dX t = π t X t ( µdt + νdW t + σdB t ) − c t X t dt. (41)As before, admissible strategies are given by F -progressively measurable R × R + -valuedprocesses ( π, c ) satisfying E (cid:82) T ( π t + c t ) dt < ∞ , and every admissible strategy results in astrictly positive wealth process.Because this is a mean field game with common noise B , the mean field equilibriumcondition will involve conditional means given B . Intuitively, because the interaction be-tween the agents occurs through the (geometric) average over the whole population, weexpect some kind of a law of large numbers and asymptotic independence between theagents as n → ∞ . Due to the presence of common noise, any asymptotic independencebetween the agents must be conditional on the common noise B , and we refer to [7, 8] formore thorough and precise treatments of mean field games with common noise. In otherwords, the population average wealth and consumption processes should be adapted to thecomplete filtration F B = ( F Bt ) t ∈ [0 ,T ] generated by the common noise B . Now, supposethat the representative agent knows that the geometric mean wealth and consumption ofthe (continuum of) other agents are governed by some F B -adapted processes X and Γ,respectively. Then, the objective of the representative agent is to maximize the expectedpayoff sup ( π,c ) ∈A MF E (cid:20)(cid:90) T U (cid:16) c t X t (Γ t X t ) − θ ; δ (cid:17) dt + (cid:15)U (cid:16) X T X − θT ; δ (cid:17)(cid:21) . (42) ONSUMPTION AND INVESTMENT UNDER RELATIVE PERFORMANCE CRITERIA 13
In equilibrium, the optimal ( π ∗ , c ∗ ) for this problem should lead to exp E [log X t | F Bt ] = X t and exp E [log c ∗ t | F Bt ] = Γ t , where we note that exp E [log( · )] is the continuous analogue ofgeometric mean. We formalize this discussion in the following definition: Definition 3.1.
Let ( π ∗ , c ∗ ) be admissible strategies, and consider the F B -adapted pro-cesses X t := exp E [log X ∗ t |F Bt ] and Γ t = exp E [log c ∗ t |F Bt ], where ( X ∗ t ) t ∈ [0 ,T ] is the wealthprocess in (41) corresponding to the strategy ( π ∗ , c ∗ ). We say that ( π ∗ , c ∗ ) is a meanfield equilibrium if ( π ∗ , c ∗ ) is optimal for the optimization problem (42) corresponding tothis choice of X and Γ. We call ( π ∗ , c ∗ ) a strong equilibrium if π ∗ is constant (and thus F -measurable) and if ( c ∗ t ) t ∈ [0 ,T ] is continuous and F -measurable.Because F is the σ -field generated by the type vector, to say that a strategy is F -measurable simply means that it depends on the type vector only, not on the Brownianmotions or wealth process. We may now state a theorem which explains the precise sensein which the n → ∞ limiting strategies computed above can be viewed as the equilibriumoutcome of a mean field game. Theorem 3.2.
Assume that a.s. δ > , θ ∈ [0 , , (cid:15) > , µ > , σ ≥ , ν ≥ , and σ + ν > . Define ( φ, ψ, β, ρ, λ ) as in (35) – (38) , and assume all of the expectations thereinare finite. Then there is a unique strong equilibrium ( π ∗ , c ∗ ) , and it takes the form givenby (39) and (40) . Corollary 3.3. (Single Stock) Assume that ( µ, ν, σ ) is deterministic with ν = 0 and µ, σ > . Then β defined in (36) can be simplified to β = µ σ (1 − δ ) (cid:18) − θθ crit (cid:19) (cid:18) δ − θθ crit ( δ − (cid:19) , where θ crit := 1 + E [ θ ( δ − E [ δ ] , and the optimal investment simplifies to π ∗ = (cid:18) δ − θθ crit ( δ − (cid:19) µσ . We omit the proof, because it closely parallels the proofs of Theorem 2.2 and [21,Theorem 3.6]. The main idea, as in the proof of Theorem 2.2, is to identify the dynamics ofthe process X t = exp E [log X t | F Bt ], when X is subject to F -measurable strategies ( π, c ),with π time-independent. The representative agent’s optimization problem can then be castas a (tractable) stochastic control problem over the two-dimensional state process ( X, X ).4.
Discussion of the equilibrium
We now discuss the interpretation of the equilibria computed in the previous sectionsand the nature of the dependence on the various model parameters. First, notice that ourresult is consistent with Samuelson’s [26], in the sense that the investment strategy weobtain is the same as in the model without consumption, derived in the previous work [21].More generally, the investment strategy π ∗ does not depend on the relative importance thatthe agents give to terminal wealth versus consumption, quantified by (cid:15) in our model. Withthis in mind, we refer to [21] for the discussion on the investment strategy, and we focusthe rest of the discussion here on the consumption strategy.We further limit the discussion of the equilibrium to the mean field case, for which theequilibrium consumption policy is given by (40), as the equilibrium in the n -agent gamehas essentially the same structure but more complicated formulas. Moreover, we restrict our attention mostly to the single stock case of Corollary 3.3, which is again more tractablebut already quite rich.From the expression for c ∗ t , we can distinguish three regimes of consumption behavior.The optimal consumption is necessarily a monotone function of time t , and a quick compu-tation shows that it is increasing when β < λ , decreasing when β > λ and constant when β = λ . Recalling the form of the wealth process X in (41), we see that the expected rateof return of wealth, ddt E [log X t | F ], is also a monotone function of time, with the oppo-site monotonicity of the consumption policy. (Note that conditioning on F is equivalent toconditioning on the representative agent’s type.) See Figure 1 for some typical consumptionpolicies. Figure 1.
Equilibrium consumption c ∗ t versus t for various values of δ .The parameters are µ = 5, σ = 1, (cid:15) = 1, E [log( (cid:15) δ )] = 0, E [ θ ( δ − . E [ δ ] = 3, θ = 0 .
8, and T = 1. Note that the final consumption c ∗ T = λ doesnot depend on δ .Recall that (cid:15) captures the relative importance that an agent gives to terminal wealthcompared to consumption. Note that λ → ∞ as (cid:15) →
0, and in particular we have β < λ forsmall (cid:15) . As discussed above, this means the agent aims for a decreasing rate of growth ofwealth and an increasing rate of consumption. This is natural, because a small (cid:15) indicatesthe agent’s lack of interest in terminal wealth, which drives X ( t ) toward zero as t → T (asthere is no bequest in our model). That is, for (cid:15) sufficiently small, the agent dis-investsafter some time in order to consume more. In fact, the agent may even begin dis-investingimmediately if π ∗ < c ∗ (0). On the contrary, if (cid:15) is large, the agent is more concernedwith terminal wealth than consumption and will thus decrease her consumption over time.Indeed, λ is decreasing in (cid:15) , so for large (cid:15) we have β > λ .We now turn to the key question of the impact of an agent’s competitiveness and risktolerance on her consumption behavior. To simplify the discussion, let us assume henceforththat no agent has a preference between her utility of wealth or utility of consumption; thatis, (cid:15) = 1 and E [log( (cid:15) − δ )] = 0, which in particular implies λ = 1. Note that if θ = 0, thenwe recover the classical Merton solution without competition, with β = µ σ δ (1 − δ ) and π ∗ = δµ/σ . For general θ , we may still rewrite β and π ∗ in an analogous manner as β = µ σ δ eff (1 − δ eff ) , π ∗ = δ eff µ/σ , ONSUMPTION AND INVESTMENT UNDER RELATIVE PERFORMANCE CRITERIA 15 where we define the effective risk tolerance parameter δ eff := (cid:18) − θθ crit (cid:19) δ + θθ crit . In other words, in the face of competition, the agent behaves like a Merton investor/consumerbut with a different risk tolerance parameter. We can interpret δ eff as a weighted averageof the agent’s own risk tolerance δ and the critical log-investor case δ log = 1, with theweight determined by the agent’s competitiveness. Take note, however, that the range ofthe weight θ/θ crit is [0 , ∞ ) and δ eff can be negative, so we should avoid interpreting δ eff tooliterally as a risk tolerance parameter.Let us investigate in more detail what distinguishes between agents who decrease versusincrease their consumption over time, and let us continue to assume that (cid:15) ≡ λ ≡
1) for all agents. As discussed above, an agent increases her consumption over time ifand only if β < λ = 1. If 8 σ > µ , then we always have β <
1, because δ eff (1 − δ eff ) ≤ / δ eff . So assume instead that 8 σ < µ . Then, because β is a quadratic function of δ (if all other parameters are held fixed), we may find δ ∗± such that β > ⇐⇒ δ ∈ ( δ ∗− , δ ∗ + ) . That is, the agent decreases consumption over time if δ ∗− < δ < δ ∗ + , increases consumptionover time if δ / ∈ ( δ ∗− , δ ∗ + ), and consumes at a constant rate if δ ∈ { δ ∗− , δ ∗ + } . Precisely, thesetwo values are δ ∗± := 1 + 12 (cid:32) θ/θ crit − ± (cid:112) − σ /µ (cid:12)(cid:12) θ/θ crit − (cid:12)(cid:12) (cid:33) . (If θ = θ crit , then β = 0, so let us assume θ (cid:54) = θ crit .) This explains the non-monotonicity in δ of the equilibrium consumption strategy, as well as the wave-like shape of the curve of c ∗ t versus δ and θ pictured in Figure 2. In the classical Merton problem with no competition,recovered by setting θ = 0, the endpoints become δ ∗± = (1 ± (cid:112) − σ /µ ), both of whichare less than 1; in this case only risk averse ( δ <
1) agents decrease their consumption overtime. In contrast, in the competitive case θ >
0, the interval ( δ − , δ + ) may lie above orbelow 1 depending on the sign of 1 − θ/θ crit .On the one hand, suppose the agent is less competitive than the critical value, or θ < θ crit . Then δ ∗− < δ ∗ + <
1, and we have seen that the agent will decrease her rateof consumption over time only if her risk tolerance lies within the range ( δ ∗− , δ ∗ + ). As wewould expect, a relatively uncompetitive agent behaves similarly to a Merton investor inthis respect. On the other hand, if the agent is more competitive than the critical value, or θ > θ crit , then 1 < δ ∗− < δ ∗ + . This means that even highly risk tolerant agents may decreaseconsumption over time. Noting that δ eff decreases with θ , one interpretation is as follows:Increasing θ exposes an agent to relative performance pressures, which is itself a sourceof risk, and to offset this additional risk the agent behaves like a Merton investor with asmaller risk tolerance parameter.Note that a highly competitive agent, with θ > θ crit , behaves in a sense opposite to howthey would if θ = 0. Indeed, when θ > θ crit , the effective risk tolerance δ eff is less than 1 if δ > δ <
1. The agent effectively switches to the other side of thecritical risk tolerance δ log = 1.We have seen by now how, with other parameters held fixed, the consumption policymay depend non-monotonically on the risk tolerance δ , with an intermediate range of risktolerance parameters ( δ ∗− , δ ∗ + ) in which agents decrease consumption over time. Similarly,with other parameters held fixed, consumption can exhibit the same non-monotonicities asa function of θ , with an intermediate range ( θ ∗− , θ ∗ + ) in which agents decrease consumption Figure 2.
Optimal consumption versus ( δ, θ ) at time
T /
2. The parametersare µ = 5, σ = 1, (cid:15) = 1, E [log (cid:15) ] = 0, E [ θ ( δ − . E [ δ ] = 5, and T = 1.over time. See Figure 3 for a depiction of the range of ( δ, θ ) parameters leading agents todecrease versus increase consumption over time. Figure 3.
Consumption regime versus ( δ, θ ). An agent with ( δ, θ ) lyinginside (resp. outside) the shaded region decreases (resp. increases) consump-tion rate over time. Agents on the boundary consume at a constant rate.Note there is a small unshaded wedge near the origin. The parameters are µ = 5, σ = 1, (cid:15) = 1, E [log (cid:15) ] = 0, E [ θ ( δ − . E [ δ ] = 5. Here θ crit = 0 . References [1] A.B. Abel. Asset prices under habit formation and catching up with the Joneses. Technical report,National Bureau of Economic Research, 1990.[2] M. Anthropelos, T. Geng, and T. Zariphopoulou. Competitive investment strategies under forwardperformance criteria. 2017. In preparation.
ONSUMPTION AND INVESTMENT UNDER RELATIVE PERFORMANCE CRITERIA 17 [3] S. Basak and D. Makarov. Competition among portfolio managers and asset specialization. Availableat SSRN: https://ssrn.com/abstract=1563567, 2015.[4] E. Bayraktar, J. Cvitanic, and Y. Zhang. Large tournament games. 2018.[5] J. Bielagk, A. Lionnet, and G. dos Reis. Equilibrium pricing under relative performance concerns.
SIAMJournal on Financial Mathematics , 8(1):435–482, 2017.[6] R. Carmona, M. Cerenzia, and A.Z. Palmer. The Dyson game. arXiv preprint arXiv:1808.02464 , 2018.[7] R. Carmona and F. Delarue.
Probabilistic Theory of Mean Field Games with Applications I-II . Springer,2018.[8] R. Carmona, F. Delarue, and D. Lacker. Mean field games with common noise.
The Annals of Proba-bility , 44(6):3740–3803, 2016.[9] Y.L. Chan and L. Kogan. Catching up with the Joneses: Heterogeneous preferences and the dynamicsof asset prices.
Journal of Political Economy , 110(6):1255–1285, 2002.[10] P.M. DeMarzo, R. Kaniel, and I. Kremer. Diversification as a public good: Community effects inportfolio choice.
The Journal of Finance , 59(4):1677–1716, 2004.[11] G. dos Reis and T. Zariphopoulou. Forward utilities and mean-field games under relative performanceconcerns. Work in progress, 2019.[12] G.-E. Espinosa and N. Touzi. Optimal investment under relative performance concerns.
MathematicalFinance , 25(2):221–257, 2015.[13] W. Fleming and H.M. Soner.
Controlled Markov processes and viscosity solutions , volume 25. SpringerScience & Business Media, 2006.[14] C. Frei and G. dos Reis. A financial market with interacting investors: does an equilibrium exist?
Mathematics and financial economics , 4(3):161–182, 2011.[15] J. Gal´ı. Keeping up with the joneses: Consumption externalities, portfolio choice, and asset prices.
Journal of Money, Credit and Banking , 26(1):1–8, 1994.[16] O. Gu´eant, J.-M. Lasry, and P.-L. Lions. Mean field games and applications. In
Paris-Princeton lectureson mathematical finance 2010 , pages 205–266. Springer, 2011.[17] M. Huang, R.P. Malham´e, and P.E. Caines. Large population stochastic dynamic games: closed-loopMcKean-Vlasov systems and the Nash certainty equivalence principle.
Communications in Information& Systems , 6(3):221–252, 2006.[18] I. Karatzas, J.P. Lehoczky, S.P. Sethi, and S.E. Shreve. Explicit solution of a general consump-tion/investment problem. In
Optimal Consumption and Investment with Bankruptcy , pages 21–56.Springer, 1997.[19] I. Karatzas, J.P. Lehoczky, and S.E. Shreve. Optimal portfolio and consumption decisions for a smallinvestor on a finite horizon.
SIAM journal on control and optimization , 25(6):1557–1586, 1987.[20] I. Karatzas and G. ˇZitkovi´c. Optimal consumption from investment and random endowment in incom-plete semimartingale markets.
The Annals of Probability , 31(4):1821–1858, 2003.[21] D. Lacker and T. Zariphopoulou. Mean field and n-agent games for optimal investment under relativeperformance criteria. arXiv preprint arXiv:1703.07685 , 2017.[22] J.-M. Lasry and P.-L. Lions. Mean field games.
Japanese Journal of Mathematics , 2(1):229–260, 2007.[23] R.C. Merton. Lifetime portfolio selection under uncertainty: The continuous-time case.
The review ofEconomics and Statistics , pages 247–257, 1969.[24] R.C. Merton. Optimum consumption and portfolio rules in a continuous-time model. In
StochasticOptimization Models in Finance , pages 621–661. Elsevier, 1975.[25] H. Pham.
Continuous-time stochastic control and optimization with financial applications , volume 61.Springer Science & Business Media, 2009.[26] P.A. Samuelson. Lifetime portfolio selection by dynamic stochastic programming. In
Stochastic Opti-mization Models in Finance , pages 517–524. Elsevier, 1975.[27] S. Sethi and M. Taksar. Infinite-horizon investment consumption model with a nonterminal bankruptcy.
Journal of optimization theory and applications , 74(2):333–346, 1992.[28] E.R. Sirri and P. Tufano. Costly search and mutual fund flows.
The journal of finance , 53(5):1589–1622,1998.[29] L.-H. Sun. Systemic risk and interbank lending. arXiv preprint arXiv:1611.06672 , 2016.
Industrial Engineering & Operations Research, Columbia University, New York, NY.
E-mail address : [email protected] Industrial Engineering & Operations Research, Columbia University, New York, NY.
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