Efficiency of equilibria in games with random payoffs
EEFFICIENCY OF EQUILIBRIA IN RANDOM BINARY GAMES
MATTEO QUATTROPANI † AND MARCO SCARSINI
Abstract.
We consider normal-form games with n players and two strategies for each playerwhere the payoffs are Bernoulli random variables. We define the average social utility associatedto a strategy profile as the sum of the payoffs of all players divided by n . We assume that payoffvectors corresponding to different profiles are i.i.d., and the payoffs within the same profile areconditionally independent given some underlying random parameter. Under these conditions weexamine the asymptotic behavior of the average social utilities that correspond to the optimum, tothe best and to the worst pure Nash equilibrium. We perform a detailed analysis of some particularcases showing that these random quantities converge, as n → ∞ , to some function of the models’parameters. Moreover, we show that these functions exhibit some interesting phase-transitionphenomena. Introduction
The concept of Nash equilibrium (NE) is central in game theory. Nash (1950, 1951) proved thatevery finite game admits mixed Nash equilibria (MNE). In general, pure Nash equilibria (PNE)may fail to exist. Given that the concept of pure Nash equilibrium is epistemically more clearlyunderstood than the one of MNE, it is important to understand how rare it is to have gameswithout PNE. One way to address the problem is to consider games in normal form whose payoffsare random. In a random game the number of PNE is also a random variable, whose distributionis interesting to study.It is known that this distribution depends on the assumptions made on the distribution of therandom payoffs. The simplest case that has been considered in the literature deals with i.i.d. payoffshaving a continuous distribution function. This implies that ties happen with probability zero. Evenin this simple case, although it is easy to compute the expected number of PNE, the characterizationof their exact distribution is non-trivial. Asymptotic results exist as either the number of playersor the number of strategies for each player diverge. In both cases the number of PNE converges toa Poisson distribution with parameter .Generalizations of the simple case can be achieved either by removing the assumptions that allpayoffs are independent or by allowing for discontinuities in their distribution functions, or both.In both cases the number of PNE diverges and some central limit theorem (CLT) holds.To the best of our knowledge, the literature on this topic has focused on the distribution of thenumber of PNE but not on their social utility (SU), i.e., the sum of the payoffs of each player. Theissue of efficiency of equilibria and its measure has received an attention for more than a centuryand, at the end of the last millennium, has led to the definition of the price of anarchy (PoA) as apessimistic measure of inefficiency (Koutsoupias and Papadimitriou, 1999, Papadimitriou, 2001a),followed by the price of stability (PoS) as its optimistic counterpart (Schulz and Stier Moses, 2003,Anshelevich et al., 2008). The PoA is the ratio of the optimum SU over the SU of the worstequilibrium. The PoS is the ratio of the optimum SU over the SU of the best equilibrium. It isinteresting to study how these three quantities behave in a random game.1.1. Our contribution.
We consider a model with n players and two strategies for each player.Payoffs are assumed to be random. To be more precise the payoff vectors corresponding to each a r X i v : . [ m a t h . P R ] J u l trategy profile are assumed to be i.i.d. and payoffs within the same strategy profile s to be con-ditionally i.i.d. Bernoulli random variables, given a parameter Φ( s ) distributed according to theprobability law π on [0 , . A model with a similar dependence structure was considered in Rinottand Scarsini (2000), but there the payoffs have a Gaussian distribution.We will study the asymptotic behavior of the average social utilities (ASUs) in this game as n → ∞ . In particular, we focus our analysis on the optimal ASU, on the ASUs of the best, theworst, and the typical PNE.As a preliminary step, we will consider the asymptotic behavior of the random number of PNE.We consider three relevant cases for the measure π . First we look at the case where the supportof π is the whole interval [0 , and we show that the asymptotic behavior of the number of PNEdoes not depend on π . Moreover we show that in this case the asymptotic behavior of the ASU ofthe optimum, and the best equilibrium coincide and have maximal ASU, i.e., equal to 1. On theother hand, we show that efficiency of the worst PNE depends on π only through its mean.The same analysis is performed for the case in which π is the Dirac mass at p ∈ (0 , , whichcorresponds to i.i.d. payoffs.Finally we deal with a model where the dependence within the profile depends on a single pa-rameter q and perform the same asymptotic analysis as a function of p and q .For each of these models we analyze the behavior of the best and worst equilibria as a functionof the relevant parameters, showing some interesting irregularities.The techniques we use in this paper are standard in the probabilistic literature, and amountmostly to first and second moment analysis, large deviations and calculus. Nonetheless, a refinedanalysis of a perturbation of the large deviation rate of binomial random variables is required toprovide precise asymptotic results on the phase-transition mentioned in the abstract.1.2. Related literature.
The distribution of the number of PNE in games with random payoffshas been studied for a number of years. Many papers assume the random payoffs to be i.i.d. froma continuous distribution. Under this hypothesis, several papers studied the asymptotic behaviorof random games, as the number of strategies grows. For instance, Goldman (1957) showed thatin zero-sum two-person games the probability of having a PNE goes to zero. He also briefly dealtwith the case of payoffs with a Bernoulli distribution. Goldberg et al. (1968) studied general two-person games and showed that the probability of having at least one PNE converges to − e − .Dresher (1970) generalized this result to the case of an arbitrary finite number of players. Otherpapers have looked at the asymptotic distribution of the number of PNE, again when the numberof strategies diverges. Powers (1990) showed that, when the number of strategies of at least twoplayers goes to infinity, the distribution of the number of PNE converges to a Poisson (1) . She thencompared the case of continuous and discontinuous distributions. Stanford (1995) derived an exactformula for the distribution of PNE in random games and obtained the result in Powers (1990) as acorollary. Stanford (1996) dealt with the case of two-person symmetric games and obtained Poissonconvergence for the number of both symmetric and asymmetric PNE.In all the above models, the expected number of PNE is in fact 1. Under different hypotheses,this expected number diverges. For instance, Stanford (1997, 1999) showed that this is the case forgames with vector payoffs and for games of common interest, respectively. Rinott and Scarsini (2000)weakened the hypothesis of i.i.d. payoffs; that is, they assumed that payoff vectors correspondingto different strategy profiles are i.i.d., but they allowed some dependence within the same payoffvector. In this setting, they proved asymptotic results when either the number of players or thenumber of strategies diverges. More precisely, if each payoff vector has a multinormal exchangeabledistribution with correlation coefficient ρ , then, if ρ is positive, the number of PNE diverges and acentral limit theorem holds. Raič (2003) used Chen-Stein method to bound the distance betweenthe distribution of the normalized number of PNE and a normal distribution. His result is very eneral, since it does not assume continuity of the payoff distributions. Takahashi (2008) consideredthe distribution of the number of PNE in a random game with two players, conditionally on thegame having nondecreasing best-response functions. This assumption greatly increases the expectednumber of PNE. Daskalakis et al. (2011) extended the framework of games with random payoffsto graphical games. Strategy profiles are vertices of a graph and players’ strategies are binary, likein our model. Moreover, their payoff depends only on their strategy and the strategies of theirneighbors. The authors studied how the structure of the graph affects existence of PNE and theyexamined both deterministic and random graphs. Amiet et al. (2019) showed that in games with n players and two actions for each player, the key quantity that determines the behavior of thenumber of PNE is the probability that two different payoffs assume the same value. They thenstudied the behavior of best-response dynamics in random games.The issue of solution concepts in games with random payoffs has been explored by various authorsin different directions. For instance, Cohen (1998) studied the probability that Nash equilibria (bothpure and mixed) in a finite random game maximize the sum of the players’ payoffs. This bears somerelation with what we do in this paper.The fact that selfish behavior of agents produces inefficiencies goes back at least to Pigou (1920)and has been studied in various fashions in the economic literature. Measuring inefficiency ofequilibria in games has attracted the interest of the algorithmic-game-theory community aroundthe change of the millennium. Efficiency of equilibria is typically measured using either the PoAor the PoS. The PoA, i.e., the ratio of the optimum SU over the SU of the worst equilibrium,was introduced by Koutsoupias and Papadimitriou (1999) and given this name by Papadimitriou(2001b). The PoS, i.e., the ratio of the optimum SU over the SU of the best equilibrium, wasintroduced by Schulz and Stier Moses (2003) and given this name by Anshelevich et al. (2008). Thereader is referred for instance to Roughgarden and Tardos (2007) for the basic concepts related toinefficiency of equilibria.1.2.1. Connections with random Constraint Satisfaction Problems (CSP) and partially-oriented per-colation.
A CSP amounts to find an initialization for a set of n variables taking value in a finitealphabet, say { , } , subject to a certain number of constraints. Example of problems in this classare classical in the computer science literature, e.g. SAT, graph coloring, independent set, etc. See,among others, Coja-Oghlan (2009), Mezard and Montanari (2009). Clearly, a binary game can bephrased as a CSP by considering pure Nash equilibria as the solution concept.Random CSP have attracted at lot of attention in the physics community, where a number ofdeep conjectures on the behavior of the solution set have been developed, and only part of themhave been recently rigorously proved by mathematicians (see, e.g., Achlioptas and Peres (2004),Abbe and Montanari (2014), Ding et al. (2015)). Given a law on the space of instances of a CSP,the first problem lies in the analysis of the size of the solution set, which is a random subset of { , } n .In Amiet et al. (2019) the authors noticed that a random binary game can be phrased as amarked partially oriented percolation on the hypercube. Strategy profiles represent vertices of thehypercube, each vertex has an array mark, which corresponds to the utilities of the players underthe corresponding strategy profile. We place an oriented arc between two profiles if and only if theyare neighbors in the hypercube and the mark in the differing coordinate is strictly larger in thearrival vertex. In this framework, the set of Nash equilibria coincide with the set of vertices havingout-degree equal to zero, i.e., sinks.In the physicists’ language, in Amiet et al. (2019) the authors computed the quenched free-energy of the model, see Eq. (2.20). In this work we consider a closely related CSP, in which we enlargethe set of constraints: a “solution” is a pure Nash equilibrium with a certain social utility. In thepercolation representation of the problem, we aim at controlling the number of sinks with a given um of the entries in the mark. We will see how this additional constraint affects the free-energyand, in general, we refine the analysis of the solution set in Amiet et al. (2019) under the binary-payoff assumption. We stress that, by our analysis, in the case of binary random games a “vanilla”second-moment argument (see Achlioptas and Peres (2004)) is sufficient to control the quenchedfree-energy of the random CSP.1.3. Organization of the paper.
The rest of the paper is organized as follows. Section 2 is devotedto a second moment analysis under no assumption on the distribution F . In section Sections 3–5we present a precise analysis of the efficiency of equilibria for three different specific choices for F .Finally, Section 6 is devoted to proofs. 2. General model
We consider a game with n players. We use the symbol [ n ] for the set of players. Each player canchoose one action in { , } . Then the set Σ of strategy profiles is the Cartesian product × i ∈ [ n ] { , } .As usual, the symbol ⊕ will denote the binary XOR operator, defined as(2.1) ⊕ ⊕ , ⊕ ⊕ . Therefore, ⊕ -adding changes one action into the other. Moreover, for every s = ( s , . . . , s n ) ∈ Σ we let the symbol s − i denote the strategy profile in which the action of player i is unspecified, sothat s = ( s − i , s i ) , for all s ∈ Σ and i ∈ [ n ] .Let NE denote the set of Nash equilibria, i.e.,(2.2) NE := { s ∈ Σ | u i ( s ) ≥ u i ( s − i , s i ⊕ , ∀ i ∈ [ n ] } . For i ∈ [ n ] , u i : Σ → R denotes player i ’s payoff function. We further assume that the payoffs are binary , in the sense that(2.3) u i ( s ) ∈ { , } , ∀ i ∈ [ n ] , ∀ s ∈ Σ . We will refer to such games with the name of binary games .We will be interested in the behavior of the following quantities: social utility (SU) SU ( s ) := (cid:88) i ∈ [ n ] u i ( s ) , (2.4) average social utility (ASU) ASU ( s ) := n SU ( s ) . (2.5)In particular, we will focus on the extremes of the social utility, in the sense that we consider thefollowing objectssocial utility of the socially optimum (SO) SO ( s ) := max s ∈ Σ SU ( s ) , (2.6) social utility of the best equilibrium (BEq) Beq ( s ) := max s ∈ NE SU ( s ) , (2.7) social utility of the worst equilibrium (WEq) Weq ( s ) := min s ∈ NE SU ( s ) . (2.8)In what follows, we will consider binary games with random payoffs. More precisely, for every choiceof n ∈ N we will consider a probability measure on the set of binary games with n players as follows.Consider a random potential function , Φ : Σ → [0 , , such that(2.9) (cid:0) Φ( s ) (cid:1) s ∈ Σ , i.i.d. Φ( s ) ∼ π, for some probability measure π with supp( π ) ⊆ [0 , . Notice that considering the common-interestsgame with payoffs(2.10) u i ( s ) = Φ( s ) , ∀ i ∈ [ n ] , ∀ s ∈ Σ e have a potential game. In our model, instead, we consider a discrete perturbation of the potentialstructure, in the sense that we use the potential Φ just to model dependences between payoffs ofdifferent players under the same profile. More precisely, given the value of the potential at a givenprofile s , i.e. Φ( s ) , the utility of the players are n i.i.d. Bernoulli random variables of parameter Φ( s ) . Moreover, we assume independence of payoffs vectors under different profiles.We will call p = E [Φ( s )] . Notice that, marginally,(2.11) P ( u i ( s ) = 1) = p. Eq. (2.11) implies that the marginal distribution of the payoffs does not depend on the specificchoice of π , but only on its expectation.In the following section we will present precise results concerning three specific but significantexamples: • Fully supported potential: π is fully supported in the whole interval [0 , . • Dirac potential: π is the Dirac mass at p . Notice that in this case the sequence ( u i ( s )) i ∈ [ n ] , s ∈ Σ is i.i.d.. For this reason we will refer to this model as the independent case . • Dichotomous potential:
For some q ∈ [0 , , π is the convex combination of two Diracmasses, i.e., for every I ⊆ [0 , ,(2.12) π ( I ) = (1 − p ) δ (1 − q ) p ( I ) + pδ q +(1 − q ) p ( I ) . Notice that if q = 0 we are back to the independent case, while if q = 1 we have a.s. acommon-interests game.We stress that an interpolation of the techniques used in what follows are in principle sufficient tostudy the general model with arbitrary distribution of the potential. In fact, in this first section wewill investigate the first and the second moment of the set of solutions, i.e., the set of equilibria,without any assumption on the measure π . As we will see, the expected number of equilibriagrows exponentially with the number of players, regardless of the specific form of π . Moreover, theindependence of the payoffs across different profiles is sufficient to ensure that the random numberof equilibria is well approximated by its expectation. Proposition 1.
For any probability measure π with mean p we have (2.13) E [ | NE | ] = (cid:90) (2 − p (1 − x )) n dπ ( x ) , and (2.14) lim inf n →∞ n log E [ | NE | ] ≥ log 32 . Proposition 2.
For any probability measure π we have (2.15) lim n →∞ E [ | NE | ] E [ | NE | ] = 1 . The next corollary follows immediately by Chebyshev’s inequality and Propositions 1 and 2.
Corollary 1.
For any probability measure π , if exists some c ∈ [log(3 / , log(2)] such that (2.16) lim n →∞ n log E [ | NE | ] = c, then (2.17) n log | NE | P −→ c. or the independent model Eq. (2.13) reads(2.18) E [ | NE | ] = (1 + α ( p )) n , where(2.19) α ( p ) := p + (1 − p ) ≥ . In fact, the independent model is a particular instance of the more general one introduced in Amietet al. (2019), where the authors show that(2.20) n log | NE | P −→ log(1 + α ( p )) . In fact, the analogue of Eq. (2.20) can be proved for other models, as it is stated by Eq. (2.17).The same phenomenon occurs for the set of equilibria with a certain social utility, as soon as theexpected size of this set grows exponentially in n . More precisely, if we call(2.21) W k = { s ∈ Σ | SU ( s ) = k } , Z k = { s ∈ NE | SU ( s ) = k } ⊂ W k , the following proposition holds. Proposition 3.
Let Q = | Z k | or Q = | W k | for some k ∈ { , . . . , n } . Then, for any probabilitymeasure π for which (2.22) lim n →∞ n log E [ Q ] = c > , we have (2.23) lim n →∞ E [ Q ] E [ Q ] = 1 , and, consequently, (2.24) n log Q P −→ c. Fully supported potential
In this section we focus on the case in which π is fully supported in [0 , . We will show that,under this assumption, the number of equilibria grows at the maximal possible rate. Theorem 1 (Number of equilibria and typical efficiency) . If π is fully supported in [0 , , then (3.1) lim n →∞ n log | NE | P −→ log 2 . Moreover, if (cid:99) NE ε is the set of equilibria having average social utility greater than − ε , then for all ε > , (3.2) lim n →∞ | (cid:99) NE ε || NE | P −→ . Notice that when u i ( s ) = 1 for all i then the profile s is automatically a pure equilibrium. Onthe other hand, if the social utility of s is xn for some x ∈ (0 , , then the probability that s is anequilibrium is exponentially small, with a rate depending only on x and p , more precisely,(3.3) P ( s ∈ NE | SU ( s ) = xn ) = (cid:104) (1 − p ) (1 − x ) (cid:105) n . The rationale underlying Theorem 1 is that—given that π is fully supported—for all ε > thereexists a fraction δ of strategy profiles with average social utility larger than − ε . For those profiles,the probability of being an equilibrium has a small exponential cost. In other words, the proof of heorem 1 shows that in this framework best equilibria are optimal and have average social utilityequal to . Moreover, most of the equilibria share this property.On the other hand, for the behavior of the worst equilibria, we need to analyze the exponentialrate in Eq. (3.3). The following Theorem 2 shows that, if p < , then arbitrary bad equilibria exist.On the other hand, if p is sufficiently large, the worst equilibria have a typical average social utility,which depends only on p . Theorem 2. If π is fully supported in [0 , , then (3.4) n ( SO , Beq , Weq ) P −→ (1 , , h ( p )) , where h : (0 , → [0 , is the non-decreasing continuous function defined as (3.5) h ( p ) := (cid:40) if p ≤ , log(2(1 − p ))log(1 − p ) if p > . Figure 1.
Plot of the function h ( p ) defined in Eq. (3.5). Independent payoffs
In this section we analyze the independent model which, as mentioned above, is a particularinstance of the model in Amiet et al. (2019). In this framework, the study of the behavior of therandom variable SO is somehow classical in the probabilistic literature. In fact, the latter can bethought of as the maximum of n independent random variables with law Bin ( n, p ) . Therefore, theanalysis of SO relies on the study of the large deviation rate of a sequence of Binomial trials andhas been performed in details, e.g., in Durrett (1979). Clearly, when one focuses on Beq ( Weq )the analysis is more complicated, due to the fact that dependencies arise when restricting themaximization (minimization) to the random domain NE . In this context, the behavior of Beq and
Weq can be determined by a precise analysis of the interplay of two different factors: the exponentialcost needed to have a large average social utility (i.e., equal to some x > p ) and the exponentialcost of being an equilibrium given an average social utility equal to x . Such a competition realizesin the phenomena described in the forthcoming Theorems 3 and 4. Theorem 3 (Convergence) . There exists three explicit functions (4.1) x opt , x beq , x weq : [0 , → [0 , , such that (4.2) n (cid:0) SO , Beq , Weq (cid:1) P −→ (cid:0) x opt ( p ) , x beq ( p ) , x weq ( p ) (cid:1) . .0 0.2 0.4 0.6 0.8 1.00.20.40.60.81.0 Figure 2.
Numerical approximation of the functions defined in Eq. (4.1). In blue: x opt ( p ) .In orange: x beq ( p ) . In green: x weq ( p ) . The limit quantities for SO , Beq and
Weq —seen as a functions of p —display an interesting be-havior. In particular, there exists some threshold for the value of p before/after which the functionsstay constant. Theorem 4 (Phase transitions) . The functions x opt and x beq are both increasing on the interval (0 , / and are identically equal to on the interval [1 / , .The function x weq is identically on the interval (0 , − √ / and is increasing on the interval [1 − √ / , . We stress that in this model efficiency can always be “nearly achieved” at equilibrium, in thesense that the ratio x opt /x beq is near 1 for all the value of p ∈ (0 , , see Fig. 2.Notice that by choosing p = 1 / we are considering the uniform measure on the space of binarygames with n players. In other words, properties that hold with high probability in the model withpotential distribution π ( · ) = δ / ( · ) are shared by a fraction of binary games that approaches as n grows to infinity. Therefore, if p = , we can rephrase Theorem 4 as a counting problem and obtainthe following result. Corollary 2.
For all ε > consider the set G n of all the binary games with n players and the subset (cid:101) G n,ε of binary games having SO , Beq ∈ [1 − ε, and Weq ∈ [ x weq (1 / − ε, x weq (1 /
2) + ε ] . Then,for all ε > , lim n →∞ | (cid:101) G n,ε ||G n | = 1 . Roughly, Corollary 2 states that asymptotically almost every binary game has
PoS ≈ and PoA ≈ . . Underlying dichotomous potential
We now consider the dichotomous potential case, which can be equivalently defined as follows.For every s ∈ Σ consider an auxiliary sequence of random variables ( X ( s )) s ∈ Σ i.i.d. Bern ( p ) , asequence ( R i ( s )) i ∈ [ n ] , s ∈ Σ of i.i.d. Bern ( q ) and, finally, a sequence ( Y i ( s )) i ∈ [ n ] , s ∈ Σ of i.i.d. Bern ( p ) .Moreover, we assume all these sequences to be independent. We then define the game as follows:(5.1) u i ( s ) = (cid:40) X ( s ) if R i ( s ) = 1 Y i ( s ) if R i ( s ) = 0 , ∀ s ∈ Σ , ∀ i ∈ [ n ] . oughly, with probability q the payoff of player i under s copies the random potential X ( s ) , while,with the remaining probability, it is an independent Bern ( p ) random variable.Our next theorem shows that the exponential size of | NE | increases monotonically with the corre-lation parameter q . Moreover, asymptotically the efficiency of almost every equilibrium correspondsto the value that optimizes the competition of the exponential costs mentioned in Section 4. We willgive an explicit expression for this value and we will show that it is increasing in q for every fixed p ∈ (0 , . Therefore, the presence of correlation not only increases the number of Nash equilibria,but also their typical efficiency. Theorem 5 (Number of equilibria and typical efficiency) . For all ( p, q ) ∈ [0 , × [0 , , we have (5.2) n log | NE | P −→ log (cid:0) α ( p ) + 2 qp (1 − p ) (cid:1) . Moreover, given any ε > and defining (5.3) (cid:99) NE ε = (cid:8) s ∈ NE : (cid:12)(cid:12) n SU ( s ) − x + ( p, q ) (cid:12)(cid:12) < ε (cid:9) , where (5.4) x + ( p, q ) = q + p (1 − q )1 − p (1 − q ) + p (1 − q ) , we have (5.5) | (cid:99) NE ε || NE | P −→ . Remark 1.
Since x + (1 / ,
0) = 2 / , an immediate consequence of Theorem 5 is that, in the samespirit of Corollary 2, a uniformly sampled equilibrium in a uniformly random binary game has anaverage social utility of / . We will now establish the analogue of Theorem 3 for the general model with q ≥ . Notice thatthe limit functions in this case depend on the interplay of the two parameters p and q . Theorem 6 (Convergence) . There exists three functions (5.6) x opt , x beq , x weq : (0 , × [0 , → [0 , , such that (5.7) n (cid:0) SO , Beq , Weq (cid:1) P −→ (cid:0) x opt ( p, q ) , x beq ( p, q ) , x weq ( p, q ) (cid:1) . Given Eq. (5.7), it is natural to analyze the behavior of the limit quantities as functions of thetwo parameters p and q , in the same vein of Theorem 4. In this case, we will fix the parameter p ∈ (0 , and vary the correlation parameter q ∈ [0 , . We now show that these functions exhibitdifferent kinds of irregularity depending on the choice of p . Theorem 7 (Phase transitions) . For all ( p, q ) ∈ (0 , × [0 , the function x weq ( p, q ) has the followingproperties (i) If p ∈ (cid:2) , − √ (cid:3) , then x weq ( p, q ) = 0 for every q ∈ [0 , . (ii) If p ∈ (cid:2) − √ , (cid:3) , then x weq ( p, · ) is continuous in [0 , . Moreover, calling (5.8) ρ ( p ) := 4 p − p − − p ) p ,x weq ( p, · ) ∈ C (cid:0) (0 , ρ ( p )) (cid:1) with (5.9) ddq x weq ( p, q ) < , .0 0.2 0.4 0.6 0.8 1.00.20.40.60.81.0 0.0 0.2 0.4 0.6 0.8 1.00.20.40.60.81.00.0 0.2 0.4 0.6 0.8 1.00.20.40.60.81.0 Figure 3.
The figures represent numerical approximations of x opt ( p, q ) (blue), x beq ( p, q ) (orange) and x weq ( p, q ) (green) as function of q , when p = 0 . , . , . and . , respectively.In third picture, the vertical segment around x ≈ . is the discontinuous jump mentionedin Theorem 7. and x weq ( q, p ) = 0 for every q > ρ ( p ) . (iii) If p > , x weq ( p, q ) > for every q ∈ [0 , . (iv) There exist some critical p c ∈ (1 / , ( p c ≈ . ) such that • If p ∈ ( p c , , then x weq ( p, q ) is continuous for q ∈ [0 , . • If p ∈ (1 / , p c ) , then x weq ( p, q ) is continuous for q ∈ [0 , \ { q ∗ ( p ) } , where (5.10) q ∗ ( p ) := 1 − p + 2 p p . Moreover, (5.11) lim q ↑ q ∗ ( p ) x weq ( p, q ) < lim q ↓ q ∗ ( p ) x weq ( p, q ) . As we remarked above, in the independent model the efficiency is approximatively achieved atequilibrium, in the sense that, for all p ∈ (0 , , x opt and x beq are not far apart. The followingproposition shows that the same is true in the model with dependent payoffs. Proposition 4.
For all p, q ∈ (0 , the functions x opt ( p, q ) and x beq ( p, q ) satisfy, (5.12) x opt ( p, q ) = x opt ( q + (1 − q ) p, , x beq ( p, q ) ≥ x beq ( q + (1 − q ) p, . Proofs
In this section we present the main proofs of the results of Sections 2–5. In particular, in Sec-tion 6.1 we deal with the moments results in Section 2. In Section 6.2 we prove the convergencesin Theorems 3 and 6. In Section 6.2 we prove the phase transition outlined in Theorems 4 and 7.Finally, in Section 6.3 we will prove the results in Section 3. e will adopt the following notation. For every n ∈ N , (cid:16) Ω ( n ) , P ( n ) π (cid:17) denotes the probabilityspace introduced above, where π is the probability law of the potential and n is number of players.Since we are interested in the asymptotic scenario in which the number of players grows to infinity,we will usually drop the dependence on n in the notation. Moreover, when the choice of π isclear by the context, we will also drop the dependence on F . We say that a sequence of realrandom variables ( X n ) n ∈ N in the product probability space × n ∈ N (cid:0) Ω ( n ) , P ( n ) π (cid:1) converges to (cid:96) ∈ R inprobability (denoted by X n P −→ (cid:96) ), if(6.1) ∀ ε > , lim n →∞ P ( n ) π (cid:0)(cid:12)(cid:12) X n − (cid:96) (cid:12)(cid:12) < ε (cid:1) = 1 . Moments estimates.
We start the subsection by proving Proposition 1, namely the expectedsize of the set of equilibria.
Proof of Proposition 1.
We start by computing the probability that a given profile s is an equilib-rium by conditioning on the value of the potential at s , namely P ( s ∈ NE ) = (cid:90) P ( s ∈ NE | Φ( s ) = x ) dF ( x ) (6.2) = (cid:90) [1 − p (1 − x )] n dF ( x ) , (6.3)(6.4)where we used the fact that P ( s ∈ NE | Φ( s ) = x ) = P ( (cid:54) ∃ i ∈ [ n ] s.t. u i ( s ) = 0 , u i ( s − i , s i ⊕
1) = 1) (6.5) = (cid:89) i ∈ [ n ] (cid:0) − P ( u i ( s ) = 0 , u i ( s − i , s i ⊕
1) = 1) (cid:1) (6.6) = [1 − P ( u ( s ) = 0 , u ( s − , s ⊕
1) = 1)] n (6.7) = [1 − p (1 − x )] n . (6.8)Therefore, by the linearity of the expectation,(6.9) E [ | NE | ] = 2 n P ( s ∈ NE ) , from which the first part of the thesis follows. Notice that, regardless of the specific choice of p theexpected number of equilibria grows exponentially in n . In fact, by the fact that π has mean p ,there exists some ε > such that P (Φ( s ) ≥ p ) > ε . Then E [ | NE | ] = (cid:90) (2 − p (1 − x )) n dπ ( x ) (6.10) ≥ (cid:90) p (2 − p (1 − x )) n dπ ( x ) (6.11) ≥ ε (2 − p (1 − p )) n (6.12) ≥ ε (cid:18) (cid:19) n . (6.13)Hence,(6.14) lim inf n →∞ n log E [ | NE | ] ≥ log 32 . (cid:3) roof of Propositions 2 and 3. We now aim at computing the second moment of the quantities | Z k | , | W k | and | NE | . We start the proof with the analysis of the second moment of | W k | , which is easierto compute. Indeed, for every distinct s , s (cid:48) ∈ Σ , thanks to the independence of the payoffs vectoracross profile, we have(6.15) P ( s , s (cid:48) ∈ W k ) = P ( s ∈ W k ) . Therefore, E [ | W k | ] ≤ E [ | W k | ] = (cid:88) s ∈ Σ (cid:88) s (cid:48) ∈ Σ P ( s , s (cid:48) ∈ W k ) (6.16) = (cid:88) s ∈ Σ P ( s ∈ W k ) + (cid:88) s ∈ Σ (cid:88) s (cid:48) (cid:54) = s P ( s ∈ W k ) (6.17) ≤ E [ | W k | ] + E [ | W k | ] . (6.18)In particular, if there exists some subset of values k ∈ [ n ] and some ε > for which(6.19) lim inf n →∞ n log E [ | W k | ] ≥ ε, then Eqs. (6.16) and (6.19) ensure that for those k ’s the following asymptotic estimate holds(6.20) E [ | W k | ] E [ | W k | ] ≥ − ε ) n . The above argument fails if the W k is replaced by its subset Z k . This is due to the fact that(6.21) P ( s , s (cid:48) ∈ Z k ) (cid:54) = P ( s ∈ Z k ) . Since the proof is identical for both | Z k | and | NE | , we will prove the lemma using | Z k | . The prooffor | NE | is similar.We claim that for every s , s (cid:48) differing in at least three strategies, the events { s ∈ Z k } and { s (cid:48) ∈ Z k } are independent. We notice that the event { s ∈ Z k } is measurable with respect to the σ -field(6.22) σ (cid:0) { u i ( s ) } , { u i ( s − i , s i ⊕ } : i ∈ [ n ]) . We remark that in the independent case, i.e., π ( · ) = δ p ( · ) , if s (cid:48) differs from s in at least two strategieswe have that the events { s ∈ Z k } and { s (cid:48) ∈ Z k } are measurable with respect to independent σ -fields,hence they are independent. On the other hand, in the general case, the events { s ∈ Z k } , { s (cid:48) ∈ Z k } are still measurable with respect to the σ -fields of the type in Eq. (6.22), nonetheless, if s , s (cid:48) differ ina only one or two strategies, such σ -fields are not independent. Notice that, in particular, { s ∈ Z k } is measurable with respect to the σ -field generated by the complete information about the payoffsof all the players in the neighboring profiles, i.e.,(6.23) σ (cid:0) { u i ( s ) } , { u (( s − i , s i ⊕ } : i ∈ [ n ] (cid:1) . Clearly, if s , s (cid:48) differs in more than two strategies, then they are measurable with respect to inde-pendent σ -fields, hence are independent.We now want to upper bound the probability of the event { s , s (cid:48) ∈ Z k } when the two profiles s and s (cid:48) differs in only one or two strategies. We start by analyzing the case in which there exists aunique i ∈ [ n ] such that(6.24) s (cid:48) = (cid:0) s − i , s i ⊕ (cid:1) . hen, P (cid:0) s , s (cid:48) ∈ Z k (cid:1) = P ( s ∈ Z k ) P (cid:0) s (cid:48) ∈ Z k | s ∈ Z k (cid:1) (6.25) = P ( s ∈ Z k ) P (cid:0) s (cid:48) ∈ Z k | u i ( s (cid:48) ) ≤ u i ( s ) (cid:1) = P ( s ∈ Z k ) P ( s (cid:48) ∈ Z k ∩ u i ( s (cid:48) ) ≤ u i ( s )) P ( u i ( s (cid:48) ) ≤ u i ( s ))= 11 − p (1 − p ) P ( s ∈ Z k ) P (cid:0) s (cid:48) ∈ Z k ∩ u i ( s (cid:48) ) = u i ( s ) (cid:1) . Notice that the probability of the intersection must satisfy, uniformly in k , P ( n ) (cid:0) s ∈ Z k ∩ u i ( s (cid:48) ) = u i ( s ) (cid:1) = (1 − p ) P ( n − ( s ∈ Z k )+ k> · p · P ( n − ( s ∈ Z k − ) = Θ (cid:16) P ( n ) ( s ∈ Z k ) (cid:17) . Hence, we can conclude that for all distinct s , s (cid:48) ∈ Σ differing in exactly one strategy, uniformly in k ,(6.26) P (cid:0) s , s (cid:48) ∈ Z k (cid:1) = Θ (cid:16) P ( s ∈ Z k ) (cid:17) . Let us now consider the case in which s and s (cid:48) are at distance . In other words, assume that thereare two distinct players i and j , such that(6.27) s (cid:48) = (cid:0) s − ij , s i ⊕ , s j ⊕ (cid:1) . Consider also the intermediate strategies(6.28) s (cid:48)(cid:48) := (cid:0) s − j , s j ⊕ (cid:1) = (cid:0) s (cid:48)− i , s (cid:48) i ⊕ (cid:1) , s (cid:48)(cid:48)(cid:48) := (cid:0) s − i , s i ⊕ (cid:1) = (cid:0) s (cid:48)− j , s (cid:48) j ⊕ (cid:1) . Arguing as in Eq. (6.25) we get P (cid:0) s , s (cid:48) ∈ Z k (cid:1) = P ( s ∈ Z k ) P (cid:0) s (cid:48) ∈ Z k | s ∈ Z k (cid:1) (6.29) = P ( s ∈ Z k ) P (cid:0) s (cid:48) ∈ Z k | u i ( s ) ≥ u i ( s (cid:48)(cid:48)(cid:48) ) ∩ u j ( s ) ≥ u j ( s (cid:48)(cid:48) ) (cid:1) =Θ (cid:0) P ( s ∈ Z k ) (cid:1) . (6.30)We can now compute the second moment. Denote by N (cid:96) ( s ) the set of strategy profiles differingfrom s in exactly (cid:96) coordinates. By the asymptotic estimates in Eqs. (6.25) and (6.29) we canconclude that there exists some constant C = C ( p ) > , such that E [ | Z k | ] = (cid:88) s ∈ Σ (cid:88) s (cid:48) ∈ Σ P (cid:0) s , s (cid:48) ∈ Z k (cid:1) = (cid:88) s ∈ Σ P ( s ∈ Z k ) + (cid:88) s (cid:48) ∈N ( s ) P ( s , s (cid:48) ∈ Z k ) + (cid:88) s (cid:48) ∈N ( s ) P ( s , s (cid:48) ∈ Z k ) + (cid:88) s (cid:48) ∈N ≥ ( s ) P ( s ∈ Z k ) ≤ E | Z k | + C · n · ( n + n ( n − P ( s ∈ Z k ) + 2 n · (2 n − n − n ( n − − P ( s ∈ Z k ) = E | Z k | + (1 + o (1))2 n P ( s ∈ Z k ) = E | Z k | + (1 + o (1)) E [ | Z k | ] . Therefore, if lim inf n →∞ n log E | Z k | > ε for some ε > , (cid:3) (6.31) E [ | Z k | ] E [ | Z k | ] ≤ C · n · n P ( s ∈ Z k ) n · P ( s ∈ Z k ) + 1 E | Z k | ≤ ε ) n .2. Proof of the convergence.
Notice that Theorem 3 is a particular case of Theorem 6, with q = 0 . In this section we will use a unified approach, showing directly Theorem 6 and obtainingTheorem 3 by taking q = 0 . The first lemma deals with the expected size of the sets in Eq. (2.21).We remind to the reader that the entropy of a Bernoulli ( x ) is defined as H : (0 , → (0 , ∞ ) ,where(6.32) H ( x ) := − x log( x ) − (1 − x ) log(1 − x ) . The following definitions are needed.
Definition 1.
Consider the following bounded analytic functions • The function f W : (0 , → [0 , is defined as f W ( p, x ) := 2 p x (1 − p ) − x e H ( x ) . • The function f Z : (0 , → [0 , is defined as f Z ( p, x ) := 2 p x (1 − p ) − x ) e H ( x ) . • The function g W : (0 , → [0 , is defined as g W ( p, q, x ) := max { f W ( q + (1 − q ) p, x ) , f W ((1 − q ) p, x ) } . • The function g + Z : (0 , → [0 , is defined as g + Z ( p, q, x ) = f W ( q + (1 − q ) p, x )(1 − p ) − x . • The function g − Z : (0 , → [0 , is defined as g − Z ( p, q, x ) = f W ((1 − q ) p, x )(1 − p ) − x . • The function g Z : (0 , → [0 , is defined as g Z ( p, q, x ) := (1 − p ) − x g W ( p, q, x ) = max (cid:8) g − Z ( p, q, x ) , g + Z ( p, q, x ) (cid:9) . Remark 2.
Notice that, for all ( p, x ) ∈ (0 , g W ( p, , x ) = f W ( p, x ) , g − Z ( p, , x ) = g + Z ( p, , x ) = g Z ( p, , x ) = f Z ( p, x ) . Moreover, for all p ∈ (0 , , the functions defined in Definition 1 admit the limits x ↑ and x ↓ ,see Lemmas 1 and 2. Therefore, for all p ∈ (0 , we can extend the functions in Definition 1, asfunctions of the second variable, to continuous functions in [0 , . The forthcoming Lemmas 1 and 2 establish some easy facts about the behavior of the functionsdefined in Definition 1, which can be checked by direct computation.
Lemma 1.
The functions f W and f Z have the following properties: (i) For every p ∈ (0 , (6.33) ∂∂x log f W ( p, x ) = log( η ( p, x )) , where (6.34) η ( p, x ) := p − p · − xx . Hence, fixed any p ∈ (0 , (6.35) ∂∂x f W ( p, x ) = 0 ⇐⇒ x = p. Moreover, (6.36) f W ( p, p ) = 2 . ii) For every x ∈ (0 , (6.37) ∂∂p log f W ( p, x ) = τ ( p, x ) , where (6.38) τ ( p, x ) := x − pp (1 − p ) . Hence, fixed any x ∈ (0 , (6.39) ∂∂p f W ( p, x ) = 0 ⇐⇒ p = x. (iii) For every p ∈ (0 , (6.40) lim x ↑ f W ( p, x ) = 2 p, lim x ↓ f W ( p, x ) = 2 − p. (iv) For every p, x ∈ (0 , (6.41) f W ( p, x ) > f Z ( p, x ) . (v) For every p ∈ (0 , (6.42) ∂∂x log f Z ( p, x ) = log ( β ( p, x )) where (6.43) β ( p, x ) := p (1 − p ) · − xx hence, fixed any p ∈ (0 , (6.44) ∂∂x f Z ( p, x ) = 0 ⇐⇒ x = (cid:98) x ( p ) := p − p + p , moreover (6.45) f Z ( p, (cid:98) x ( p )) = 1 + p + (1 − p ) =: 1 + α ( p ) . (vi) For every x ∈ (0 , (6.46) ∂∂p log f Z ( p, x ) = υ ( p, x ) , where (6.47) υ ( p, x ) := x − p (2 − x ) p (1 − p ) . Hence, fixed any x ∈ (0 , (6.48) ∂∂p f Z ( p, x ) = 0 ⇐⇒ p = x − x . (vii) For every p ∈ (0 , (6.49) lim x ↓ f Z ( p, x ) = 2(1 − p ) , lim x ↑ f Z ( p, x ) = 2 p. Lemma 2.
The functions g + Z ( p, q, x ) and g − Z ( p, q, x ) , defined in Definition 1, have the followingproperties: a) For every p, q ∈ (0 , ∂∂x log g + Z ( p, q, x ) = log( β + ( p, q, x )) , ∂∂x log g − Z ( p, q, x ) = log( β − ( p, q, x )) (6.50) where (6.51) β + ( p, q, x ) := q + (1 − q ) p (1 − p ) (1 − q ) · − xx , β − ( p, q, x ) := (1 − q ) p (1 − p )(1 − p (1 − q )) · − xx . (b) For all p, q ∈ (0 , there exists two points (6.52) x + ( p, q ) := q + p (1 − q )1 − p (1 − q ) + p (1 − q ) , x − ( p, q ) := p (1 − q )1 − p + p (1 − q ) such that (6.53) ∂∂x g ± Z ( x ) > if x < x ± ( p, q ) , = 0 if x = x ± ( p, q ) ,< if x > x ± ( p, q ) . (c) For every p, q, x ∈ (0 , ∂∂q log g + Z ( p, q, x ) = q + p ( q −
1) + x ( q − p ( q −
1) + q ) , (6.54) ∂∂q log g − Z ( p, q, x ) = p ( q −
1) + x ( q − p ( q −
1) + 1) . Moreover, (6.55) ∂∂q g ± Z ( p, q, x ) > if x < υ ± ( p, x ) , = 0 if x = υ ± ( p, x ) ,< if x > υ ± ( p, x ) , where (6.56) υ − ( p, x ) := p − xp , υ + ( p, x ) := x − p − p . (d) For every p, q ∈ (0 , (6.57) g + Z ( p, q, x ) < g − Z ( p, q, x ) if x < γ ( p, q ) ,> g − Z ( p, q, x ) if x = γ ( p, q ) ,> g − Z ( p, q, x ) if x > γ ( p, q ) , where, (6.58) γ ( p, q ) := log θ ( p, q )log θ ( p, q ) , (6.59) θ ( p, q ) := (1 − p )(1 − q )1 − p (1 − q ) , θ ( p, q ) := θ ( p, q ) p (1 − q ) q − p (1 − q ) . (e) For every p, q ∈ (0 , (6.60) lim x ↓ g − Z ( p, q, x ) = 2(1 − p )(1 − p (1 − q )) , lim x ↓ g + Z ( p, q, x ) = 2(1 − p ) (1 − q ) , and (6.61) lim x ↑ g − Z ( p, q, x ) = 2 p (1 − q ) , lim x ↑ g + Z ( p, q, x ) = 2 (cid:0) q + p (1 − q ) (cid:1) . emark 3. Notice that if q = 0 we are back to the independent case, indeed g ± Z ( p, , x ) = f Z ( p, x ) for every p, x ∈ (0 , and β ± ( p, , x ) = p (1 − p ) · − xx , x ± ( p,
0) = p − p + p g ± Z ( p, , x ± ( p, α ( p ) . Lemma 3.
For all ( p, q ) ∈ (0 , × [0 , , ε > , k ∈ [ εn, (1 − ε ) n ] we have (6.62) n log E | W k | ∼ log g W (cid:0) p, q, kn (cid:1) , n log E | Z k | ∼ log g Z (cid:0) p, q, kn (cid:1) . Proof of Lemma 3.
By independence of payoffs under different strategy profile, we have E [ | W k | ] =2 n P ( s ∈ W k )=2 n · [ p · P ( Bin ( n, q + (1 − q ) p ) = k ) + (1 − p ) · P ( Bin ( n, (1 − q ) p ) = k )]= p · n · (cid:18)(cid:18) nk (cid:19) ( q + (1 − q ) p ) k (1 − q − (1 − q ) p ) n − k (cid:19) ++ (1 − p ) · n · (cid:18)(cid:18) nk (cid:19) ((1 − q ) p ) k (1 − (1 − q ) p ) n − k (cid:19) , where, to get the second equality, we conditioned on the outcome of X ( s ) , defined at the beginningof Section 5.Fix ε > , and pick some k ∈ [ εn, (1 + ε ) n ] . Let x = kn , and notice that using the asymptoticapproximation(6.63) (cid:18) nxn (cid:19) = e (1+ o (1)) H ( x ) n , we can estimate E [ | W k | ] = p · (cid:104) (1 + o (1))2 e H ( x ) ( q + (1 − q ) p ) x (1 − q − (1 − q ) p ) − x (cid:105) n + (6.64) + (1 − p ) · (cid:104) (1 + o (1))2 e H ( x ) ((1 − q ) p ) x (1 − (1 − q ) p ) − x (cid:105) n = p · [(1 + o (1)) f W ( q + (1 − q ) p, x )] n + (1 − p ) · [(1 + o (1)) f W ((1 − q ) p, x )] n . Notice that the convex coefficients p and − p are absorbed in the (1+ o (1)) error within the squaredbrackets. Hence, if k ∈ [ εn, (1 − ε ) n ] for any ε > , E [ | W k | ] =[(1 + o (1)) max { f W ( q + (1 − q ) p, kn ) , f W ((1 − q ) p ) } ] n . By taking the logarithm and normalizing,(6.65) n log E | W k | ∼ max { log f W ( q + (1 − q ) p, kn ) , log f W ((1 − q ) p, kn ) } . We now compute the expected size of Z k . By conditioning on the social utility of the profile s ∈ Σ E [ | Z k | ] =2 n P ( s ∈ Z k )=2 n P ( s ∈ NE | s ∈ W k ) P ( s ∈ W k ) . Notice that, conditioning on any ω ∈ { s ∈ W k } , the probability that s is an equilibrium is exactlythe probability of the event(6.66) { u i ( s − i , s i ⊕
1) = 0 , ∀ i s.t. u i ( s ) = 0 } . By the independence across i ’s in the events in Eq. (6.66), we conclude that(6.67) P (cid:92) i : u i ( s )=0 u i ( s − i , s i ⊕
1) = 0 (cid:12)(cid:12) s ∈ W k = (1 − p ) n − k . ence, using the same argument as in Eq. (6.64), E [ | Z k | ] =2 n · (1 − p ) n − k · [ p · P ( Bin ( n, q + (1 − q ) p ) = k ) + (1 − p ) · P ( Bin ( n, (1 − q ) p ) = k )]= (cid:2) (1 + o (1)) max { g − Z ( p, q, kn ) , g + Z ( p, q, kn ) } (cid:3) n . Therefore, if k ∈ [ εn, (1 − ε ) n ] for any ε > , n log E [ | Z k | ] ∼ max { log g − Z ( p, q, kn ) , log g + Z ( p, q, kn ) } . (cid:3) We can use the result in Lemma 3 to show Theorem 5, which controls the number and typicalefficiency of equilibria.
Proof of Theorem 5.
In order to compute the expectation of | NE | it is sufficient to notice that P ( s ∈ NE ) = p P ( s ∈ NE | X ( s ) = 1) + (1 − p ) P ( s ∈ NE | X ( s ) = 0) (6.68) = p (1 − p (1 − q )(1 − p )) n + (1 − p ) (1 − p ( q + (1 − q )(1 − p )) n . (6.69)Therefore E [ | NE | ] = 2 n P ( s ∈ NE ) = p [2 (1 − p (1 − q )(1 − p ))] n (1 + o (1)) (6.70) = p [1 + α ( p ) + 2 qp (1 − p )] n (6.71) =Ω ((3 / n ) (6.72)Therefore, by Proposition 2, | NE | / E [ | NE | ] P −→ , and by Eq. (6.70) we get Eq. (5.2). At this point,in order to show the validity of Eq. (5.5) it is sufficient to notice that for every sufficiently small ε > < n log (cid:16) E [ | NE \ (cid:99) NE ε | ] (cid:17) < log (1 + α ( p ) + 2 qp (1 − p )) and again by Proposition 2, | (cid:99) NE ε | / E [ | (cid:99) NE ε | ] P −→ . (cid:3) Notice that if there exists some δ > such that g W ( p, q, k/n ) ≥ δ then the expectation E | W k | = ω (1) , while if g W ( p, q, k/n ) ≤ − δ then the expectation E | W k | = o (1) . The main ideaof the proof of Theorem 6 goes as follows. If for some triple ( p, q, x ) we have g Z ( p, q, x ) < , thenby Markov’s inequality we can infer that asymptotically there are no Nash equilibria with averagesocial utility x + o (1) . On the other hand, if g Z ( p, q, x ) > , the set of equilibria with averagesocial utility x + o (1) is not empty; this will be proved by the control on the second moments inProposition 3.We start by using Lemma 3 to control the probability that the set Z k is empty. As remarkedabove, for every k that is not too close to or n , the expectation of | Z k | is exponential of rate g NE ( p, q, k/n ) ∈ [0 , . Hence, in what follows we will be interested in studying the solution of thefollowing equations(6.73) g W ( p, q, x ) = 1 and g Z ( p, q, x ) = 1 . See Figs. 4 and 5 for a representation of the functions g W ( p, q, x ) and g Z ( p, q, x ) as a function ofthe variable x for some fixed values of p and q .For instance, when q = 0 , for every given p ∈ (0 , , the smallest x for which f Z ( p, x ) ≥ is ourproxy for the ASU of the worst equilibrium, while the largest x for which f Z ( p, x ) ≥ is our proxyfor the ASU of the best equilibrium. Similarly, the optimum ASU can be obtained by looking atthe largest x for which f W ( p, x ) ≥ . We formalize this intuition in the following proposition, whichis the technical version of the more readable Theorem 6. Proposition 5.
Fix p, q ∈ (0 , × [0 , and any ε, δ > . .0 0.2 0.4 0.6 0.8 1.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.00.51.01.52.0 Figure 4.
In Blue: f Z ( p, x ) as function of x . In Orange: f W ( p, x ) as function of x . Thevalue of p in the four pictures is, respectively, p = 0 . , − √ / , . . The green line is atheight . Figure 5.
Plot of the functions g W (0 . , . , x ) (in orange) and g Z (0 . , . , x ) (in blue).The fact that the orange curve lies above the line at height one, means that the expectationof the number of strategy with SU k diverges exponentially fast, for all k ∈ [0 , n ] . On theother hand, the regions where the blue curve lies below the green line denotes values of ASUwhich will not appear within the strategies in NE . (a) Call N + ε,δ := { k ∈ [ δn, (1 − δ ) n ] : g W (cid:0) p, q, kn (cid:1) ≥ ε } , (6.74) N − ε,δ := { k ∈ [ δn, (1 − δ ) n ] : g W (cid:0) p, q, kn (cid:1) ≤ − ε } , (6.75) we have lim n →∞ P (cid:0) ∪ k ∈ N − ε,δ W k = ∅ (cid:1) = 1 , lim n →∞ P (cid:0) ∀ k ∈ N + ε,δ , W k (cid:54) = ∅ (cid:1) = 1 . (6.76)(b) Called M + ε,δ := { k ∈ [ δn, (1 − δ ) n ] : g Z (cid:0) p, q, kn (cid:1) ≥ ε } , (6.77) M − ε,δ := { k ∈ [ δn, (1 − δ ) n ] : g Z (cid:0) p, q, kn (cid:1) ≤ − ε } , (6.78) we have, lim n →∞ P (cid:0) ∪ k ∈ M − ε,δ Z k = ∅ (cid:1) = 1 , lim n →∞ P (cid:0) ∀ k ∈ M + ε,δ , Z k (cid:54) = ∅ (cid:1) = 1 . (6.79) Proof.
We prove (a); the proof of (b) is similar.Fix a pair p, q ∈ (0 , × [0 , and ε, δ > . Pick some k ∈ N − ε,δ . We have(6.80) P ( | W k | ≥ ≤ E [ | W k | ] , hence, for every sufficiently large n (6.81) n log P ( | W k | ≥ ≤ − ε, here the first is Markov inequality and the second estimate follows from Lemma 3 and holdsuniformly in k ∈ N − ε,δ . By applying the union bound we get the first limit in Eq. (6.76).On the other hand, by Proposition 3 and the second moment method, (see Alon and Spencer,2016, Ch. 4) ,(6.82) P ( W k (cid:54) = ∅ ) ≥ E [ | W k | ] E [ | W k | ] ≥ − ε ) n . We can therefore deduce the second limit in Eq. (6.76) by applying a union bound over k ∈ N + ε,δ . (cid:3) The results in Theorems 3 and 6 now follow as a corollary of Proposition 5.
Proof of Theorem 6.
The claim follows directly by Eqs. (6.76) and (6.79). We show the result forthe
Weq , since the proofs for
Beq and SO are identical. Fix ε, δ > . Notice that, by Eq. (6.77), n min s ∈ NE SU ( s ) = n min { k ∈ [ n ] : Z k (cid:54) = ∅ } ≤ n min { k ∈ M + ε,δ } =: n k ε,δ On the other hand, by Eq. (6.79), we have(6.83) n min { k ∈ [ n ] : Z k (cid:54) = ∅ } ≥ n max { k ∈ M − ε,δ | k ≤ k ε,δ } =: n k ε,δ , where we define k ε,δ = 0 if the set in its definition is empty. By continuity of g Z ( p, q, · ) and the factthat g Z ( p, q, · ) and its derivative are bounded around 0 and 1 (see Lemma 2), we have that for all η > , we can find n , ε and δ such that, for all n > n ,(6.84) (cid:12)(cid:12)(cid:12)(cid:12) n k ε,δ − n k ε,δ (cid:12)(cid:12)(cid:12)(cid:12) < η, and(6.85) (cid:12)(cid:12) n k ε,δ − inf (cid:8) x ∈ (0 ,
1) : g Z ( p, q, x ) ≥ (cid:9)(cid:12)(cid:12) < η. Hence,(6.86) lim n →∞ P (cid:18)(cid:12)(cid:12) n min s ∈ NE SU ( s ) − inf (cid:8) x ∈ (0 ,
1) : g Z ( p, q, x ) ≥ (cid:9)(cid:12)(cid:12) ≤ η (cid:19) = 1 , which concludes the proof. (cid:3) As a byproduct of the above proof we get the following characterization of the limit functionsdefined in Theorem 6. By Eq. (6.86), for all p, q ∈ (0 , × [0 , , the function x weq defined inTheorem 6 admits the following implicit representation x weq ( p, q ) = inf { x ∈ (0 ,
1) : g Z ( p, q, x ) ≥ } . (6.87)Similarly, the functions x opt and x beq admit the representations x opt ( p, q ) = sup { x ∈ (0 ,
1) : g W ( p, q, x ) ≥ } , (6.88) x beq ( p, q ) = sup { x ∈ (0 ,
1) : g Z ( p, q, x ) ≥ } . (6.89)The phase-transition in Theorem 4 is then a consequence of the analysis of the functions inEqs. (6.87) and (6.88), Eq. (6.89). Proof of Theorem 4.
We start by analyzing the function x opt : (0 , / → [0 , .By Lemma 1(iii), we have(6.90) lim x ↑ log f W ( p, x ) < . This inequality, together with the continuity of f W ( p, x ) , justifies the characterization of x opt ( p ) asthe largest solution in x ∈ (0 , of the equation(6.91) log f W ( p, x ) = 0 . n order to use implicit function theorem, we need to check that, for every fixed p ∈ (0 , / , thepartial derivative with respect to x of the function f W ( p, x ) does not vanish at the largest solutionof Eq. (6.91). Notice that x opt ( p ) > p since f W ( p, p ) = 2 . Moreover, thanks to Lemma 1(i) for every p ∈ (0 , / and x ∈ ( p, we have(6.92) η ( p, x ) < ⇒ ∂∂x log f W ( p, x ) < , where η is defined as in Eq. (6.34). Relying again on the rough estimate x opt ( p ) > p , together withLemma 1(ii), we get(6.93) τ ( p, x ) > ⇒ ∂∂p log f W ( p, x opt ( p )) > , where τ is defined as in Eq. (6.38). Therefore, by the implicit function theorem, the implicit function x opt ( p ) defined by Eq. (6.91) admits, for all p ∈ (0 , / , derivative of the form(6.94) ddp x opt ( p ) (cid:12)(cid:12) p = − ∂∂p log f W ( p, x ) (cid:12)(cid:12) p ,x opt ( p ) ∂∂x log f W ( p, x ) (cid:12)(cid:12) p ,x opt ( p ) > . We proceed similarly for x beq ( p ) in the same interval (0 , / . By Lemma 1(vii)(6.95) lim x ↑ log f W ( p, x ) < . Hence, by continuity of f Z ( p, x ) , the function x beq ( p ) is defined implicitly as the largest solution in (0 , of the equation(6.96) log f Z ( p, x ) = 0 . Notice that, thanks to Lemma 1(v),(6.97) ∂∂x log f Z ( p, x ) < , ∀ x > (cid:98) x ( p ) := p − p + p . Since(6.98) log f Z ( p, (cid:98) x ( p )) = log(1 + α ( p )) > , it holds that x beq ( p ) > (cid:98) x ( p ) . Moreover, by Lemma 1(vi), we have(6.99) ∂∂p log f Z ( p, x ) > ∀ x > p p . Since(6.100) f Z (cid:18) p, p p (cid:19) = (1 + p )(2 − p ) p +1 − > , we have(6.101) x beq > p p . In conclusion, by the implicit function theorem, the function x beq is C (0 , / and(6.102) ddp x beq ( p ) (cid:12)(cid:12) p > . The behavior of the function x opt and x beq in the interval (1 / , follows by the continuity of f W and f Z together with Lemma 1(iv) and Lemma 1(vii). e are left to show the regularity properties of the function x weq . If we restrict to p ∈ (1 −√ / , ,we can characterize x weq as the smallest solution in x ∈ (0 , of the equation in Eq. (6.96). Noticethat by Lemma 1(iii), Eq. (6.100), and the continuity of f Z ( p, · ) we deduce that(6.103) x weq ( p ) < p p . For the same reason, given Eq. (6.98), we have x weq ( p ) < p − p + p . By the converse of the inequalitiesin Eqs. (6.97) and (6.99), we obtain ∂∂p log f Z ( p, x ) < ⇐⇒ x < p p (6.104) ∂∂x f Z ( p, x ) > ⇐⇒ x < p − p + p , (6.105)So, we can apply the implicit function theorem and conclude that the function x weq ( p ) admits aderivative(6.106) ddp x weq ( p ) (cid:12)(cid:12) p > ∀ p ∈ (1 − / √ , . To conclude, we need to show that x weq is constantly zero in the interval (cid:0) , − √ (cid:1) . This is justa simple consequence of Lemma 1(vii). In fact, lim x ↑ log f W ( p, x ) > ⇐⇒ p < − √ . (cid:3) We are now going to prove the phase-transition phenomenon described in Theorem 7. We recallto the reader that here we assume p to be fixed while the moving parameter is the correlation q . Proof of Theorem 7.
By steps:(i) By Lemma 2(e) we have(6.107) lim x ↓ g − Z ( p, q, x ) = 2(1 − p )(1 − p (1 − q )) > ⇐⇒ q > ρ ( p ) . Notice that(6.108) ρ ( p ) > ⇐⇒ p > − √ . The result follows immediately(ii) We note that ρ ( p ) : (1 − / √ , / → (0 , is a bijection. Moreover, by the implicit functiontheorem, for all q ∈ (0 , ρ ( p )) , ddq ˜ x weq ( p, q ) (cid:12)(cid:12) q = − ∂∂q log g − Z ( p, q, x ) (cid:12)(cid:12) q , ˜ x weq ( p,q ) ∂∂x log g − Z ( p, q, x ) (cid:12)(cid:12) q , ˜ x weq ( p,q ) (6.109) = − p (1 − q ) − x (1 − q )(1 − p (1 − q )) log ( β − ( p, q, x )) (cid:12)(cid:12)(cid:12)(cid:12) q , ˜ x weq ( p,q ) , (6.110) where β − is defined as in Eq. (6.51). We aim at showing that the signs of numerator anddenominator in Eq. (6.110) coincide. Note that we have(6.111) log β − ( p, q, x weq ( p, q )) > , ∀ p ∈ (cid:0) − √ , (cid:1) , ∀ q > ρ ( p ) . In fact, the sign of log β − ( p, q, · ) is the sign of the partial derivative of g − Z ( p, q, x ) with respectto x , which is positive at x weq ( p, q ) . On the other hand, by definition,(6.112) (1 − q )(1 − p (1 − q )) > . ence, we are left with showing that(6.113) p (1 − q ) − x weq ( p, q ) > , ∀ p ∈ (cid:0) − √ , (cid:1) , ∀ q > ρ ( p ) . Since for p ∈ (1 − √ / , / the quantity x weq ( p, q ) is the smallest solution of the equation(6.114) log g − Z ( p, q, x ) = 0 , it is sufficient to check that(6.115) log g − Z ( p, q, p (1 − q )) > , ∀ p ∈ (cid:0) − √ , (cid:1) , ∀ q > ρ ( p ) . We can rewrite Eq. (6.115) as(6.116) log(2) + (1 − p (1 − q )) log(1 − p ) > , ∀ p ∈ (cid:0) − √ , (cid:1) , ∀ q > ρ ( p ) . Notice that the following inequality is stronger than Eq. (6.116):(6.117) log(2) + log(1 − p ) > , ∀ p ∈ (cid:0) − √ , (cid:1) , and the latter is trivially true.(iii) By the same argument used to prove (i), we have(6.118) lim x ↓ g − Z ( p, q, x ) < ⇐⇒ q < ρ ( p ) . To conclude, notice that ρ ( p ) > if p ∈ (1 / , .(iv) For every p ∈ (1 / , call q ∗ ( p ) the unique solution of the equation in q (6.119) log g − Z ( p, q, x − ( p, q )) = log (cid:0) (cid:0) p (1 − q ) + (1 − p ) (cid:1)(cid:1) = 0 . For all p ∈ (1 / , the function g − Z ( p, · , x − ( p, · )) is decreasing in q ∈ (0 , , and the uniquesolution of the equation in Eq. (6.119) is given by(6.120) q ∗ ( p ) = 1 − p + 2 p p , ∀ p ∈ (1 / , . The existence of a unique solution to Eq. (6.119) implies that, for all p ∈ (1 / , there exista unique value of q for which the maximum attained by the curve g − Z ( p, q, · ) is exactly .Having in mind the plots in Figs. 6 and 7, we are interested in understanding whether(6.121) x − ( p, q ∗ ( p )) ≶ inf { x ∈ (0 ,
1) : g + Z ( p, q ∗ ( p ) , x ) ≥ } . In order to do so, we aim at analyzing the map(6.122) p (cid:55)→ g + Z (cid:0) p, q ∗ ( p ) , x − ( p, q ∗ ( p )) (cid:1) , namely, the value assumed by the function g + Z at the point in which the function g − Z attains itsmaximum height, i.e., 1. We start by claiming that the function in Eq. (6.122) is increasing,hence there exists a unique solution in (1 / , of the equation in p (6.123) g + Z (cid:0) p, q ∗ ( p ) , x − ( p, q ∗ ( p )) (cid:1) = 1 . We define p c the solution of Eq. (6.123); numerically, p c ≈ . . As suggested by theplots in Figs. 6 and 7, we expect two different behaviors of the function x weq ( p, · ) when p ∈ (1 / , p c ) (see Fig. 6), and when p ∈ ( p c , (see Fig. 7).In order to show the monotonicity of the map in Eq. (6.122) it is sufficient to proceed byexplicit computation. In fact,(6.124) log g + Z (cid:0) p, q ∗ ( p ) , x − ( p, q ∗ ( p )) (cid:1) = log (cid:18) − p − p (cid:19) + (cid:18) − p (cid:19) log (cid:18) − (3 − p ) p (2 p − (1 − p ) (cid:19) , .0 0.2 0.4 0.6 0.8 1.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.00.51.01.52.00.0 0.2 0.4 0.6 0.8 1.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.00.51.01.52.0 Figure 6.
Plot of g Z ( p, q, x ) (blue) and g W ( p, q, x ) (orange) when p = 0 . < p c and q =0 . , . , . , . , respectively. Recall that g Z ( p, q, x ) = g − Z ( p, q, x ) on the left of the dashedline x = γ ( p, q ) , while g Z ( p, q, x ) = g + Z ( p, q, x ) on the right. Similarly g W ( p, q, x ) = f W ((1 − q ) p, x ) on the left of the dashed line, while g W ( p, q, x ) = f W ( q + (1 − q ) p, x ) on the right. Figure 7.
Plot of g Z ( p, q, x ) (blue) and g W ( p, q, x ) (orange) when p = 0 . > p c and q =0 . , . , . , . , respectively. The dashed line lies at x = γ ( p, q ) . take the derivative ddp log g + Z (cid:0) p, q ∗ ( p ) , x − ( p, q ∗ ( p )) (cid:1) = − p + 10 p − p + 3 p (2 p − p − p + 1) + 1 p log (cid:18) p − p + 1(1 − p )(2 p − (cid:19) , nd notice that, for all p ∈ (1 / , ,(6.125) ddp log g + Z (cid:0) p, q ∗ ( p ) , x − ( p, q ∗ ( p )) (cid:1) > . Therefore, there exists a unique p c ∈ (1 / , for which(6.126) g − Z (cid:0) p c , q ∗ ( p c ) , x − ( p c , q ∗ ( p c )) (cid:1) = g + Z (cid:0) p c , q ∗ ( p c ) , x − ( p c , q ∗ ( p c )) (cid:1) = 1 . Moreover, thanks to Lemma 2(d), the value of p c can be further characterized as the unique p ∈ (1 / , such that(6.127) x − ( p c , q ∗ ( p c )) = γ ( p c , q ∗ ( p c )) , where γ is defined as in Eq. (6.58). In conclusion, if p ∈ (1 / , p c ) the function x weq ( p, · ) isdiscontinuous at q ∗ ( p ) .On the other hand, if p ∈ ( p c , , the map(6.128) q (cid:55)→ inf (cid:8) x ∈ (0 ,
1) : max (cid:0) log g − Z ( p, q, x ) , log g + Z ( p, q, x ) (cid:1) = 0 (cid:9) , is continuous in [0 , . In fact, by Eq. (6.119), g − Z ( p, q, x − ( p, q )) is decreasing in q , hence, if q ∈ ( q ∗ ( p ) , , x weq ( p, q ) = inf (cid:8) x ∈ (0 ,
1) : log g + Z ( p, q, x ) = 0 (cid:9) , which is clearly continuous.Conversely, if q ∈ (0 , q ∗ ( p )) , then the function g − Z ( p, q, · ) is increasing in (0 , γ ( p, q )) , andthe function g + Z ( p, q, · ) is increasing in a neighborhood of γ ( p, q ) . Hence, Eq. (6.128) can bewritten as(6.129) q (cid:55)→ (cid:40) inf (cid:8) x ∈ (0 ,
1) : log g − Z ( p, q, x ) = 0 (cid:9) if q < q (cid:50) ( p ) , inf (cid:8) x ∈ (0 ,
1) : log g + Z ( p, q, x ) = 0 (cid:9) if q > q (cid:50) ( p ) , where q (cid:50) ( p ) solves(6.130) g − Z ( p, q (cid:50) ( p ) , γ ( p, q (cid:50) ( p ))) = 1 . By definition of γ ,(6.131) g + Z ( p, q (cid:50) ( p ) , γ ( p, q (cid:50) ( p ))) = 1 . hence, there cannot be any discontinuity when passing from the first to the second branchof Eq. (6.129). (cid:3) Proof of Proposition 4.
By Eq. (6.88) and Lemma 1, x opt ( p, q ) = sup (cid:8) x ∈ (0 ,
1) : max ( f W ((1 − q ) p, , f W ( q + (1 − q ) p, x )) ≥ (cid:9) (6.132) = sup (cid:8) x ∈ (0 ,
1) : f W ( q + (1 − q ) p, x ) ≥ (cid:9) (6.133) = x opt ( q + (1 − q ) p, . (6.134)On the other hand, by Eq. (6.89), x beq ( p, q ) = sup (cid:8) x ∈ (0 ,
1) : (1 − p ) − x max ( f W ((1 − q ) p, x ) , f W ( q + (1 − q ) p, x )) ≥ (cid:9) (6.135) = sup (cid:8) x ∈ (0 ,
1) : (1 − p ) − x f W ( q + (1 − q ) p, x ) ≥ (cid:9) (6.136) ≥ sup (cid:8) x ∈ (0 ,
1) : (1 − p ) − x f W ( p, x ) ≥ (cid:9) (6.137) = x beq ( q + (1 − q ) p, . (6.138)where the inequality above follow from Lemma 1(ii), which implies that ∀ ( q, x ) ∈ (0 , (1 − p ) − x f W ( q + (1 − q ) p, x ) ≥ (1 − p ) − x f W ( p, x ) . (cid:3) .3. Proof for the fully supported potential.
In this subsection we focus on the case in which π is fully supported in [0 , . More precisely, we will assume that for every I ⊂ [0 , with Leb ( I ) = ε there exists some δ = δ ( ε ) such that(6.139) π ( I ) ≥ δ. Proof of Theorem 1.
We start by showing a concentration result for the number of profiles with alarge potential. By Eq. (6.139), for any fixed ε > there exists some δ = δ ( ε ) > such that P (cid:0) Φ( s ) ≥ − ε (cid:1) = δ. It follows that the expected number of profiles with potential in the interval [1 − ε, is E [ |{ s : Φ( s ) ≥ − ε }| ] = δ n . (6.140)Moreover, by the Chernoff bound, for any constant γ ∈ (0 , (6.141) P ( Binomial (2 n , δ ) < (1 − γ ) δ n ) ≤ exp ( − Θ(2 n )) . Hence, considering the family of events(6.142) E ε,c := {|{ s : Φ( s ) ≥ − ε ] }| > c n } , there exists some sufficiently small constant c = c ( ε ) > for which(6.143) lim n →∞ P ( E ε,c ) = 1 . Notice now that if Φ( s ) ≥ − ε then the probability that SU ( s ) = n can be lower bounded by(6.144) P ( SU ( s ) = n | Φ( s ) ≥ − ε ) ≥ (1 − ε ) n . Therefore, E [ | Z n | ] ≥ E [ | Z n | | |{ s : Φ( s ) ≥ − ε ] }| > c n ] P ( |{ s : Φ( s ) ≥ − ε ] }| > c n ) (6.145) ≥ c (2(1 − ε )) n (6.146)By Proposition 3,(6.147) | Z n | E [ | Z n | ] P −→ , from which, by taking ε → , follows(6.148) n log ( | Z n | ) P −→ . (cid:3) Proof of Theorem 2.
Fix some s ∈ Σ and notice that(6.149) P ( s ∈ Z k ) = P ( SU ( s ) = k ) P ( s ∈ NE | SU ( s ) = k ) . It is worth noting that the quantity P ( s ∈ NE | SU ( s ) = k ) depends only on p , regardless of thespecific form of π . In fact, P ( s ∈ NE | SU ( s ) = k ) = P ( u i ( s ) = 0) n − k (6.150) = (cid:18)(cid:90) (1 − x ) dπ ( x ) (cid:19) n − k (6.151) = (1 − p ) n − k . (6.152)We notice further that, if Φ( s ) = x then, by the law of large numbers, for all ε > n →∞ P (cid:18) SU ( s ) n ∈ [ x − ε, x + ε ] (cid:12)(cid:12) Φ( s ) ∈ [ x − ε/ , x + ε/ (cid:19) = 1 . (6.153) eing π fully supported we have P (Φ( s ) ∈ [ x − ε/ , x + ε/ , (6.154)and therefore E [ | { s : Φ( s ) ∈ [ x − ε/ , x + ε/ } | ] = Θ(2 n ) , (6.155)so that by the independence of the sequence (Φ( s )) s ∈ Σ we have | { s : Φ( s ) ∈ [ x − ε/ , x + ε/ } | E [ | { s : Φ( s ) ∈ [ x − ε/ , x + ε/ } | ] P −→ . (6.156)Therefore, by Eqs. (6.153) and (6.156) we conclude that for any arbitrarily small interval [ x − ε, x + ε ] ∈ [0 , we have Θ(2 n ) profiles with such an ASU. Moreover, by Eq. (6.152), a profile with ASU x ± ε is a NE with probability (1 − p ) (1 − x + ε ) n ≤ P ( s ∈ NE | SU ( s ) ∈ [( x − ε ) n, ( x + ε ) n ]) ≤ (1 − p ) (1 − x − ε ) n . (6.157)Hence, [2(1 − p ) (1 − x + ε ) ] n ≤ E [ |{ s : s ∈ NE , SU ( s ) /n ∈ [ x − ε, x + ε ] }| ] ≤ [2(1 − p ) (1 − x − ε ) ] n . (6.158)The theorem follows by letting ε → in Eq. (6.158), by using Proposition 3 and by noting that(6.159) − p ) (1 − x ) = 1 ⇐⇒ p ≥ and x = h ( p ) . (cid:3) Acknowledgments.
Both authors are members of GNAMPA-INdAM and of COST Action GAMENET.This work was partially supported by the GNAMPA-INdAM Project 2020 “Random walks on ran-dom games” and PRIN 2017 project ALGADIMAR.
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E-mail address : [email protected] Dipartimento di Economia e Finanza, LUISS, Viale Romania 32, 00197 Roma, Italy.
E-mail address : [email protected]@luiss.it