Games and Economic Behavior

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1 Games an Economic Behavior Contents lists available at SciVerse ScienceDirect Games an Economic Behavior Fast convergence in evolutionary equilibrium selection Gabriel E. Kreinler a,,h.peytonyoung b a Nuffiel College, New Roa, Oxfor OX1 1NF, Unite Kingom b Department of Economics, University of Oxfor, Manor Roa, Oxfor OX1 3UQ, Unite Kingom article info abstract Article history: Receive 3 July 01 Available online 5 March 013 JEL classification: C7 C73 Keywors: Stochastic stability Logit learning Marov chain Convergence time Stochastic best response moels provie sharp preictions about equilibrium selection when the noise level is arbitrarily small. The ifficulty is that, when the noise is extremely small, it can tae an extremely long time for a large population to reach the stochastically stable equilibrium. An important exception arises when players interact locally in small close-nit groups; in this case convergence can be rapi for small noise an an arbitrarily large population. We show that a similar result hols when the population is fully mixe an there is no local interaction. Moreover, the expecte waiting times are comparable to those in local interaction moels. 013 Elsevier Inc. All rights reserve. 1. Stochastic stability an equilibrium selection Evolutionary moels with ranom perturbations provie a useful framewor for explaining how populations reach equilibrium from out-of-equilibrium conitions, an why some equilibria are more liely than others in the long run. Iniviuals in a large population interact with one another repeately to play a given game, an they upate their strategies base on information about what others are oing. The upating rule is usually assume to be myopic best response with a ranom component resulting from errors, unobserve utility shocs, or experiments. The long-run behavior of the resulting stochastic ynamical system can be analyze using the theory of large eviations Freilin an Wentzell, The ey iea is to examine the limiting behavior of the process when the ranom component becomes vanishingly small. This typically leas to powerful selection results, that is, the limiting ergoic istribution tens to be concentrate on particular equilibria often a unique equilibrium that are stochastically stable Foster an Young, 1990; Kanori et al., 1993; Young, A common criticism of this approach is that the waiting time to get close to the ergoic istribution may be exceptionally large. If the noise level is small an the system starts near an equilibrium that is not stochastically stable, it will remain close to this equilibrium for a very long perio of time. Inee, the time it taes to escape the initial wrong equilibrium grows exponentially in the population size when the noise is sufficiently small Ellison, 1993; Sanholm, 010. Nevertheless, this leaves open an important question: must the noise actually be close to zero in orer to obtain sharp selection results? This assumption is neee to characterize the stochastically stable states theoretically, but it coul be that at intermeiate levels of noise the selection process isplays a fairly strong bias towars the stochastically stable states. If this is so the spee of convergence coul be quite rapi. * Corresponence to: G.E. Kreinler, Department of Economics, Massachusetts Institute of Technology, 50 Memorial Drive, Cambrige, MA 014, Unite States. aresses: ge@mit.eu G.E. Kreinler, peyton.young@economics.ox.ac.u H.P. Young /$ see front matter 013 Elsevier Inc. All rights reserve.

2 40 G.E. Kreinler, H.P. Young / Games an Economic Behavior A pioneering paper by Ellison 1993 shows that this is inee the case when agents interact locally with small groups of neighbors. The paper eals with the specific case when agents are locate at the noes of a ring networ an they are line with agents within a specifie istance. The form of interaction is a symmetric coorination game. Ellison shows that the waiting time to get close to the stochastically stable equilibrium, say all- A, is boune inepenently of population size, an that its absolute magnitue may be very small. The reason that this set-up leas to fast selection is local reinforcement. When the noise is sufficiently small but not taen to zero, any small close-nit group of interconnecte agents will not tae long to aopt the action A. Once they have one this, they will continue to play A with high probability thereafter even when people outsie the group are playing the alternative action B. Since this occurs in parallel across the entire networ inepenently of population size, it oes not tae long in expectation until almost all of the close-nit groups, an hence a large proportion of the population, have switche to A. In fact this argument is quite general an applies to a variety of local networ structures an stochastic learning rules, as shown in Young 1998, 011. A separate line of wor stuies the conitions for equilibrium selection when agents myopically best respon to their available information. Sanholm 001 consiers the case when agents best respon to ranom samples of size, an shows that any 1/-ominant equilibrium is eventually reache with high probability from almost any initial conition, for sufficiently large population size. 1 In particular, when the sample size is two an interaction is given by a symmetric coorination game with a unique ris-ominant equilibrium, say A, A, this equilibrium will be reache from any initial conition in which a positive fraction of the population plays A. The intuition for this result is that for almost any population action profile, the expecte change in the proportion of A-players is strictly positive. The ey iea is to wor with the eterministic approximation of the process corresponing to an infinite population; in this setting the A-equilibrium is an almost global attractor. When the population is large but finite, the process is stochastic but well approximate by the eterministic ynamics, an convergence occurs in boune time with high probability. A similar eterministic approximation is use to analyze the iffusion of innovations in ranom networs; see López-Pintao 006 an Jacson an Yariv 007. However, these papers o not attempt to characterize the expecte waiting time to reach equilibrium. The contribution of this paper is to provie an in-epth analysis of expecte waiting times in evolutionary moels with global interaction an stochastic best response ynamics. The moel assumes a large, finite population of ientical agents repeately interacting accoring to a symmetric coorination game. Agents occasionally have the opportunity to revise their actions, an when they o they choose a perturbe best response to the istribution of actions in the population. We shall consier the case where agents now the actual istribution of actions an also the case where they only observe a finite ranom sample of actions. In the interest of analytical tractability we shall restrict attention to coorination games an logit best response ynamics Blume, 1993, The metho of analysis applies to a much broaer class of response functions, as we show in Section 7. For the sae of concreteness we shall call action A the innovation an action B the status quo an tal about the aoption rate of the innovation A starting from the status quo when everyone is playing B. We can characterize the expecte waiting time until most players have aopte A in terms of two easily interpretable parameters: the payoff gain α of the innovation relative to the status quo, an the noise level 1/β, where β is the coefficient of the logit function. The analysis procees in four steps. First we characterize the eterministic mean-fiel ynamics for given levels of α an β, assuming an infinite population. Next, we characterize the combinations of α an β such that the eterministic ynamics have a unique rest point. We then exploit the geometry of the aggregate response function to characterize the maximum length of time it taes for the process to reach a neighborhoo of the unique rest point. Finally, we apply stochastic approximation theory Benaïm an Weibull, 003 to show that the estimations for the eterministic system are vali for the case of large but finite populations. The main results are the following: i For a given payoff gain α > 0 the ynamics exhibit a phase transition with respect to the noise level: there is a critical noise level below which the waiting time is exponential in the number of agents selection is slow, an above which the waiting time is boune irrespective of the number of agents selection is fast. ii We provie an explicit estimate of the critical noise threshol, which turns out to be quite small. In particular, when 1 β > 1 selection is fast for all positive values of α. This threshol correspons to an initial error rate the probability of choosing A when everyone else is choosing B of about 1%. iii For plausible parameter values the absolute magnitues of the waiting times are very small. For example, when the innovation is 100% better than the status quo an the agents initial error rate is 5%, it taes less than 0 revisions per capita in expectation to reach an aoption rate of 99%. These results can also be interprete in terms of heterogeneity in iniviual payoffs rather than as noisy best responses. In particular, suppose that each iniviual has an iiosyncratic payoff from playing any given strategy, in aition to the payoff from the coorination game. Assume that these payoffs are fixe across time, an that they are rawn inepenently 1 An equilibrium A, A is1/-ominant if A is a best response to any sample of size that contains at least one player playing A. A number of authors have observe that when the noise in the logit response function is large enough, the eterministic mean-fiel ynamic has a unique global attractor McKelvey an Palfrey, 1995; Blume an Durlauf, 00; Sanholm, 010, Example 6... However, they i not raw out the implications for convergence times.

3 G.E. Kreinler, H.P. Young / Games an Economic Behavior across strategies an across agents from an extreme-value istribution with mean zero, parameter β an variance π 6β.In a large population the ynamics will behave as if the iniviuals were choosing via the logit process with parameter β. Consequently, the above results may be interprete as saying that even quite low levels of variance in the payoffs can lea to fast selection. The paper is organize as follows. We begin with a review of relate literature in Section. Section 3 sets up the moel, an Section 4 contains the first main result, namely the existence an estimation of a critical payoff gain when agents have full information. We erive an upper boun on the number of steps to get close to equilibrium in Section 5. Section 6 extens the results in the previous two sections to the partial information case. In Section 7 we show that the metho of analysis extens to many types of perturbe response functions in aition to the logit.. Relate literature The rate at which a coorination equilibrium becomes establishe in a large population or whether it becomes establishe at all has been stuie from a variety of perspectives. To unerstan the connections with the present paper we shall ivie the literature into several parts, epening on whether interaction is assume to be global or local, an on whether the selection ynamics are eterministic best response or stochastic noisy best response. In the latter case we shall also istinguish between those moels in which the stochastic perturbations are taen to zero low noise ynamics, an those in which the perturbations are maintaine at a positive level noisy ynamics. To fix ieas, let us consier the situation where agents interact in pairs an play a fixe symmetric pure coorination game G of the following form: A B A 1 + α, 1 + α 0, 0, α 0 B 0, 0 1, 1 We can thin of B as the status quo, of A as the innovation an of α as the payoff gain of the innovation relative to the status quo this representation is in fact without loss of generality, as we show in Section 3. Local interaction refers to the situation where agents are locate at the noes of a networ an they interact only with their neighbors. Global interaction refers to the situation where agents react to the istribution of actions in the entire population, or to a ranom sample of such actions. Virtually all of the results about waiting times an rate of convergence can be iscusse in this setting. The essential question is how long it taes to transit from the all-b equilibrium to a state where most of the agents are playing A..1. Deterministic best response, local interaction Morris 000 stuies eterministic best-response ynamics on an infinite networ. Each noe of the networ is occupie by an agent, an in each perio all agents myopically best respon to their neighbors actions. Myopic best response, either with or without perturbations, is assume throughout this literature. Morris stuies the threshol α such that, for any payoff gain α > α, there exists some finite group of initial aopters from which the innovation spreas by contagion to the entire population. He provies several characterizations of α in terms of topological properties of the networ, such as the existence of cohesive inwar looing subgroups, the uniformity of the local interaction, an the growth rate of the number of players who can be reache in steps. A particularly interesting case occurs when all egrees in a connecte networ are at most, an the payoff gain is larger than. In this case, any single initial aopter will cause the innovation to sprea by contagion to the entire population. Morris oes not aress the issue of waiting times as such, rather, he ientifies conitions that are necessary an sufficient for full aoption to occur uner best response ynamics... Noisy best response, local interaction Ellison 1993 an Young 1998, 011 stuy aoption ynamics when agents are locate at the noes of a networ. Whenever agents revise, they best respon to their neighbors actions, with some ranom error. Unlie most moels iscusse in this section, the population is finite an the selection process is stochastic. The aim of the analysis is to characterize the ergoic istribution of the process rather than to approximate it by a eterministic ynamic, an to stuy whether convergence to this istribution occurs in a reasonable amount of time. Ellison examines the case where agents best respon with a uniform probability of error choosing a non-best response, while Young focuses on the situation where agents use a logit response rule. The latter implies that the probability of maing an error ecreases as the payoff loss from maing the error increases. In both cases, the main fining is that, when the networ consists of small close-nit groups that interact mainly with each other rather than outsiers, then for small but not vanishing levels of noise the process evolves quite rapily to a state where most agents are playing A inepenently of the size of the population, an inepenently of the initial state. Montanari an Saberi 010 consier a similar situation: agents are locate at the noes of a fixe networ an they upate asynchronously using a logit response function. The authors characterize the expecte waiting time to transit from

4 4 G.E. Kreinler, H.P. Young / Games an Economic Behavior all-b to all-a as a function of population size, networ structure, an the size of the gain α. Lie Ellison an Young, Montanari an Saberi show that local clusters ten to spee up the selection process, whereas overall well-connecte networs ten to be slow. For example, selection is slow on ranom networs for small enough α > 0, while selection on smallworl networs networs where agents are mostly connecte locally but there also exist a few ranom istant lins becomes slower as the proportion of istant lins increases. These results stan in contrast with the ynamics of isease transmission, where contagion tens to be fast in well-connecte networs an slow in localize networs Anerson an May, 1991, see also Watts an Dos, 007. The analytical framewor of Montanari an Saberi iffers from that of Ellison an Young in one crucial respect however: in the former the waiting time is characterize as the noise is taen to zero, whereas in the latter the noise is hel fixe at a small but not arbitrarily small level. This ifference has important implications for the magnitue of the expecte waiting time: if the noise is extremely small, it taes an extremely long time in expectation for even one player to switch to A given that all his neighbors are playing B. Montanari an Saberi show that when the noise is vanishingly small, the expecte waiting time to reach all-a is inepenent of the population size for some types of networs an not for others. However, their metho of analyzing this issue requires that the absolute magnitue of the waiting time is very large in either case..3. Deterministic best response, global interaction an sampling A number of authors have consiere the situation where agents interact globally an choose best responses to ranom samples rawn from the population. Some of these moels analyze the resulting stochastic process for finite populations, while others focus exclusively on the mean-fiel ynamics with a continuum of agents. We have alreay mentione the pioneering wor of Sanholm 001, who observes that when all samples have size, then any 1/-ominant equilibrium becomes a global attractor except from egenerate initial conitions. In the setting of a pure coorination game, the 1/-ominant conition is equivalent to α >. 3 Sanholm s results have recently been generalize by Oyama et al. 011 to iterate 1/-ominant equilibria an settings where the sample size is itself ranom. They show that when samples of size at most are sufficiently liely, a conition they call -gooness, then the iterate 1/-equilibrium becomes an almost global attractor. López-Pintao 006 stuies a eterministic mean-fiel approximation of a large, finite system where agents are line by a ranom networ with a given egree istribution, an in each time perio agents best respon to their neighbors actions. López-Pintao proves that, for a given istribution of sample sizes, there exists a minimum threshol value of α above which, for any positive initial fraction of A-players however small, the process evolves to a state in which a positive proportion of the population plays A forever. Jacson an Yariv 007 use a similar mean-fiel approximation technique to analyze moels of innovation iffusion. They ientify two types of equilibrium aoption levels: stable levels of aoption an unstable ones tipping points. A smaller tipping point facilitates iffusion of the innovation, while a larger stable equilibrium correspons to a higher final aoption level. Jacson an Yariv erive comparative statics results on how the networ structure changes the tipping points an stable equilibria. Finally, Watts 00 an Lelarge 01 stuy eterministic best-response ynamics on large ranom graphs with a specifie egree istribution. In particular, Lelarge analyzes large but finite ranom graphs an characterizes in terms of the egree istribution the threshol value α such that, for any payoff gain α > α, with high probability, a single player who aopts A leas the process to a state in which a positive proportion of the population is playing A. In summary, the previous literature has ealt mainly with four cases: i Deterministic best response an local interaction Morris, ii Noisy best response an local interaction Ellison, Young, iii Low noise best response an both local an global interaction Montanari an Saberi, an iv Deterministic best response an global interaction with sampling Sanholm, López-Pintao, etc.. There remains the case of noisy ynamics with global interaction. We now from prior results in the literature that when the noise in the logit best response process is large enough, a single equilibrium can survive as the global attractor of the eterministic mean-fiel ynamic McKelvey an Palfrey, 1995; Blume an Durlauf, 00; Sanholm, 010, Example 6..; Hommes an Ochea, 01. It follows that convergence to any neighborhoo of that equilibrium occurs in boune time inepenently of the population size. However, this oes not aress the question of how long it taes to get close to the equilibrium. 4 The contribution of the present paper is to erive explicit estimations of the waiting time as a function of the moel parameters. These estimations show that selection occurs very rapily when the noise is fairly small but not arbitrarily small. Furthermore, the expecte number of steps to reach the mostly- A state is similar in magnitue to the number of steps to reach such a state in local interaction moels on the orer of 0 50 perios for an initial error rate 3 Morris 000 fins the same threshol for contagion to occur in infinite -regular trees. 4 One of the few papers to estimate waiting times explicitly is Shah an Shin 010. They consier a special class of potential games which oes not inclue the case consiere here, an prove that for intermeiate values of noise the time to get close to equilibrium grows slowly but is unboune in the population size.

5 G.E. Kreinler, H.P. Young / Games an Economic Behavior of about 5% where each iniviual revises once per perio in expectation. The conclusion is that selection is fast uner global as well as local interaction for realistic non-vanishing noise levels. 3. The moel Consier a large population of N agents. Each agent chooses one of two available actions, A an B. Interaction is given by a symmetric coorination game with payoff matrix A B A a, a c, B, c b, b where a > an b > c. This game has the potential function A B A a 0 B 0 b c Define the normalize potential gain associate with passing from the B, B equilibrium to the A, A equilibrium a b c α = b c 1 Without loss of generality assume that a > b c, orequivalentlyα > 0. This maes A, A the ris-ominant equilibrium; note that A, A nee not be the same as the Pareto-ominant equilibrium. Stanar results in evolutionary game theory say that the A, A equilibrium will be selecte in the long run Blume, 003; see also Kanori et al., 1993 an Young, A particular case of special interest occurs when the game is a pure coorination game with payoff matrix A B A 1 + α, 1 + α 0, 0 B 0, 0 1, 1 We can thin of B as the status quo an of A as the innovation, an in this case α > 0 is also the payoff gain of aopting the innovation relative to the status quo. The potential function in this case is proportional to the potential function in the general case, which implies that uner logit learning an a suitable rescaling of the noise parameter, the two settings are equivalent. For the rest of this paper we will wor with the game form in. Agents revise their actions in the following manner. At times t = with N, an only at these times, one agent is N ranomly inepenently over time chosen to revise his action. 5 When revising, an agent gathers information about the current state of play. We consier two possible information structures. In the full information case, revising agents now the current proportion of aopters in the entire population. In the partial information case, revising agents ranomly sample other agents from the population with replacement, an learn their current actions, where is a positive integer that is inepenent of N. After gathering information, a revising agent i calculates the fraction x of agents in his sample who are playing A, an chooses a noisy best response given by the logit moel: e β1+αx Pri chooses A x = f x; α,β= e β1+αx + e β1 x 3 where 1/β is a measure of the noise in the revision process. For convenience we will sometimes rop the epenence of f on β an simply write f x;α, oronbothα an β an write f x. Denoteε = 1 1+e β the associate error rate at zero aoption rate or the initial error rate; given the bijective corresponence between β an ε, we will use the two variables interchangeably to refer to the noise level in the system. The logit moel is one of the two moels preominantly use in the literature. The other is the uniform error moel Kanori et al., 1993; Young, 1993; Ellison, 1993, which posits that agents mae errors with a fixe probability. A characteristic feature of the logit moel is that the probability of maing an error is sensitive to the payoff ifference between choices, maing costly errors less probable; from an economic perspective, this feature is quite natural Blume, 1993, 1995, 003. Another feature of logit is that it is a smooth response, whereas in the uniform error moel an agent s ecision changes abruptly aroun the inifference point. Finally, the logit moel can also be viewe as a pure best-response to a noisy 5 An alternative revision protocol runs as follows: time is continuous, an each agent has a Poisson cloc that rings once per perio in expectation. When an agent s cloc rings the agent revises her action. It is possible to show that results in this article remain unchange uner this alternative revision protocol.

6 44 G.E. Kreinler, H.P. Young / Games an Economic Behavior payoff observation. Specifically, if the payoff shocs ɛ A an ɛ B are inepenently istribute accoring to the extremevalue istribution given by Prɛ i z = exp expβz, then this leas to the logit probabilities Broc an Durlauf, 001; Anerson et al., 199; McFaen, The revision process just escribe efines a stochastic process Γ N α,β in the full information case an Γ N α,β, in the partial information case. The states of the process are the aoption rates x N t {0, 1 N 1,...,, 1}, an by assumption N N the process starts in the all-b state, namely x N 0 = 0. We now turn to the issue of spee of convergence, measure in terms of the expecte time until a large fraction of the population aopts action A. This measure is appropriate because the probability of being in the all- A state is extremely small. Formally, for any p < 1 efine the ranom hitting time 6 T N α,β,p = { min t: x N t } p Fast selection is efine as follows. Definition 1. The family {Γ N α,β: N > 0} isplays fast selection if there exists S = Sα,β such that the expecte waiting time until a majority of agents play A uner process Γ N α,β is at most S inepenently of N, oret N α,β, 1 <S for all N. More generally, for any p < 1, Definition. The family {Γ N α,β: N > 0} isplays fast selection to p if there exists S = Sα,β,p such that the expecte waiting time until at least a fraction p of agents play A uner process Γ N α,β is at most S inepenently of N, or ET N α,β,p<s for all N. Note. When the above conitions are satisfie then we say, by a slight abuse of language, that Γ N α,β isplays fast selection, or fast selection to p. 4. Full information The following theorem establishes how much better than the status quo an innovation nees to be in orer for it to sprea quicly in the population. Specifically, fast selection can occur for any noise level as long as the payoff gain excees a certain threshol; moreover this threshol is equal to zero for intermeiate noise levels. Theorem 1. If α > hβ then Γ N α,βisplays fast selection, where { hβ = e β 1 +4 e β, β > 0, 0 <β Moreover, when β 3hence ε < 5% an α > hβ then Γ N α,βisplays fast selection to 99%. The main message of Theorem 1 is that fast selection hols in settings with global interaction. This result oes not follow from previous results in moels of local interaction Ellison, 1993; Young, 1998; Montanari an Saberi, 010. Inee, a ey component of local interaction moels is that agents interact only with a small, fixe group of neighbors, whereas here each agent observes the actions of the entire population. Theorem 1 is nevertheless reminiscent of results from moels of local interaction. For example, Young 1998 shows that for certain families of local interaction graphs selection is fast for any positive payoff gain α > 0. Theorem 1 shows that fast selection can occur for any positive payoff gain even when interaction is global, provie that β, which is equivalent to an initial error rate larger than approximately 1%. The ey iea is that for any payoff gain α 0 there exists a critical noise level β α such that Γ N α,β isplays fast selection for β<β α. 7 Intuitively, fast selection occurs when the eterministic approximation of the process Γ N α,β as the population grows large has a unique equilibrium. Equilibria of the eterministic process correspon to fixe points of the response function. For small noise levels the logit response function has three fixe points; as the noise level increases the first part of the response function lifts up an eventually the low an mile fixe points isappear. See Fig. for an illustration of this change. Similarly, for a given noise level, the low an mile fixe points isappear for sufficiently large payoff gains. This point has been note by various authors, notably Sanholm 001, 010 an Blume an Durlauf 00; the contribution of Theorem 1 is to provie a sharp estimate of the critical payoff gain. Later in the paper in Section 5 we examine the ramifications of this phenomenon for the spee of convergence. 6 The results in this paper continue to hol uner the following stronger efinition of waiting time. Given p < 1let T α,β,p be the expecte first time such that at least the proportion p has aopte an at all later times the probability is at least p that the proportion p has aopte. 7 Also, for each noise parameter β there exists a payoff gain threshol, enote h β, suchthatγ N α,β isplays fast selection for α > h β.

7 G.E. Kreinler, H.P. Young / Games an Economic Behavior Fig. 1. The critical threshol for fast selection. Simulation blue, ashe an upper boun re, soli. The x-axis is labele with the initial error rate ε; the corresponing noise level β appears in parentheses. Fig.. The response functions for payoff gain α = 100% an initial error rate ε = 3% left panel an ε = 6% right panel. The two systems have three equilibria an a unique equilibrium, respectively. In each case, the vertical ashe line correspons to the inifference point of the pure best response. Fig. 1 shows the simulate noise threshol β α blue,ashelineaswellasthebounproviebytheorem1re, soli line. 8 The x-axis isplays the initial error rate ε = 1 1+e β, an the y-axis represents payoff gains α. Note that the ifference between the error rates given by the two curves never excees about 0.5%. Theorem 1 shows that when the payoff gain is above a specific threshol, the time until a high proportion of players aopt A is boune inepenently of the population size N. Simulations reveal that for realistic parameter values the expecte waiting time can be very small. Fig. 3 shows a typical aoption path. It taes, on average, less than 0 revisions per capita until p = 99% of the population plays A, for a population size of N = 1000, with payoff gain α = 100% an initial error rate ε = 5%. More generally, Table 1 shows how the expecte waiting time epens on the population size N, the payoff gain α, the initial error rate ε, an on the target aoption level p. 9 The last column shows the limit of the waiting time as the population size tens to infinity. The main taeaway is that the absolute magnitue of the waiting times is small. We explore this effect in more etail in Section 5. Table 1 also suggests that the expecte waiting time when the population is finite is generally less than the waiting time as the population tens to infinity. We conjecture that this is ue to the increase volatility of the process when there are finitely many agents The blue, ashe line in Fig. 1 also represents the payoff gain threshol h β, while the re, soli line represents the boun hβ. Fig. 1 also shows that the function h is continuous an that h = h = 0. 9 Note that the process Γ N α,β is a birth eath process, an the expecte waiting times to achieve a specific aoption level can be compute explicitly using stanar formulas see Example 11.A.5 in Sanholm, We than an anonymous referee for pointing this out.

8 46 G.E. Kreinler, H.P. Young / Games an Economic Behavior Fig. 3. Aoption path to 99%, α = 100%, ε = 5% β 3, N = Table 1 Expecte waiting times full information. Expecte waiting time ε = 5% N = 100 N = 1000 N = 10,000 N = α = 70% a , α = 80% a Expecte waiting time ε = 10% N = 100 N = 1000 N = 10,000 N = α = 4% b ,71.40 α = 5% c N = population size, α = innovation payoff gain, ε = initial error rate. The target aoption rate is a p = 99%, b p = 50% an c p = 90%. Note that Ellison 1993 obtains surprisingly similar simulation results in the case of local learning most of the waiting times he presents lie between 10 an 50 see Fig. 1, Tables 3 an 4 in that paper. Although the results are similar, the assumptions in the two moels are very ifferent. In Ellison s moel agents are locate aroun a circle or at the noes of a lattice an interact only with close neighbors. Also, he uses the uniform error moel instea of logit learning. Finally, in his simulations the target aoption rate is p = 75%, the payoff gain is α = 100%, an he presents results for error rates ε = 1.5%,.5% an 5%. 11 Proof of Theorem 1. The proof consists of two steps. First, we show that the results hol for a eterministic approximation of the stochastic process Γ N α,β. The secon step is to show that fast selection is preserve by this approximation when N is large. We begin by efining the eterministic approximation an the concepts of equilibrium, stability an fast selection in this setting. The eterministic process is enote Γα,β an has state variable xt. The process evolves in continuous time, an xt is the aoption rate at time t. By assumption we tae x0 = 0. In the process Γ N α,β, the probability that a revising agent chooses A when the population aoption rate is x is equal to f x;α,β. Definition 3 can be rewritten as 1 f x; α,β= 1 + e β1 α+x This function epens on α an β, but it oes not epen on N. For convenience, we shall sometimes omit the epenence of f on α an/or β in the notation. We efine the eterministic ynamic by the ifferential equation 4 ẋ = f x; α,β x 5 where the ot above x enotes the time erivative. 1 An equilibrium of this system is a rest point, that is an aoption rate x satisfying ẋ = 0, which is equivalent to f x = x. Note that this equilibrium of the process Γα,β correspons to the Logit Quantal Response Equilibrium efine in McKelvey an Palfrey An equilibrium x is stable if after any small enough perturbation the process converges bac to the same equilibrium. An equilibrium is unstable if the process never converges bac to the same equilibrium after any non-trivial perturbation. Given that f is continuously ifferentiable, x is stable if an only if f x is strictly below 1. Similarly, x is unstable if an only if f x is strictly above 1. It is easy to see that there always exists a stable equilibrium. Fig. 4 plots the position of the stable an unstable rest points as the initial error rate ε varies, for α = 5%. 11 These error rates correspon to ranomization probabilities of.5%, 5% an 10% respectively. 1 Note that the expecte change in the aoption rate in the stochastic process is given by 1 f x x. N

9 G.E. Kreinler, H.P. Young / Games an Economic Behavior Fig. 4. Rest points of the eterministic process α = 5%: stable rest points blue, soli line an unstable rest points re, ashe line. The x-axis is labele with the initial error rate ε; the corresponing noise level β appears in parentheses. The efinitions of fast selection from the stochastic setting exten naturally to the eterministic case. The hitting time to reachanaoptionratep is T α,β,p = min { t: xt p } Definition 3. The process Γα,β isplays fast selection if it reaches x = 1 in finite time, that is T α,β, 1 <. Analogously, the process Γα,β exhibits fast selection to p if T α,β,p<. Remar. A necessary an sufficient conition for fast selection is that all equilibria lie strictly above 1. Similarly, fast selection to p hols if an only if all equilibria lie strictly above p. Inee, clearly if x 1 x p is an equilibrium, an x0 = 0, then xt<x for all t. Conversely, by uniform continuity the process always reaches x = 1 x = p in finite time. The following lemma shows that the eterministic process has at most three equilibria rest points. Lemma 1. For α > 0 the process Γα,βhas a unique equilibrium x H in the interval 1, 1], an this equilibrium is stable it is referre to as the high equilibrium.furthermore,thereexistatmosttwoequilibriaintheinterval[0, 1 ]. Proof. For future reference, from 4 we fin that for all x f x = βα + f x 1 f x f x = βα + f x 1 f x Note that f +α 1 = 1, hence ientity 7 implies that The function f is strictly convex below 1 + α an strictly concave above 1 + α The result now follows by inspecting the sign of f x x for x = 0, +α an We now efine an then estimate the critical payoff gain. Using the convention inf =,let h β = inf { α: Γα,βisplays fast selection } Note that f is strictly increasing in α. 13 Given α > α, if f ;α,β oes not have any equilibria in the interval [0, 1 ],then neither oes f ;α,β. It follows that Γα,β isplays fast selection for all α > h β. 13 Differentiating Eq. 4 we obtain f α x; α,β= βxf x 1 f x This quantity is positive for all x,β >0, so f is increasing in α.

10 48 G.E. Kreinler, H.P. Young / Games an Economic Behavior The estimation of the critical payoff gain h β consists of two steps. First we show that the critical payoff gain is equal to zero if an only if β high error rates. Seconly, we establish an upper boun for the critical payoff gain for β> low error rates. Claim 1. When β then h β = 0. Whenβ> then h β > 0. To establish this claim, we stuy x L α,β, the smallest equilibrium of f. The function f is strictly increasing in α on 0, 1, hencex L is strictly increasing in α. Thus, to show that h β = 0 it is sufficient to show that x L 0,β= 1. Note that f ; 0,β is symmetric in the sense that f x; 0,β+ f 1 x; 0,β = 1, an it is strictly convex on 0, 1 an strictly concave on 1, 1. Obviouslyx = 1 is an equilibrium, so the system either has a single equilibrium or three equilibria, epening on whether f 1 ; 0,β is less than or equal to 1, or strictly greater than 1, respectively. Using 6 we have f 1 ; 0,β= β so for β the system has a single equilibrium, an thus x L0,β= 1.Forβ> the system has three equilibria, an the smallest correspons to a strict own crossing. It follows that x L 0,β< 1 an for sufficiently small α > 0 it still hols that x L α,β< 1. This implies that h β > 0. When β> the critical payoff gain threshol is non-trivial, namely for small payoff gains the system has two equilibria smaller than 1. At the critical payoff gain h β, there is a unique equilibrium smaller than 1, namely the tangency point x between the function f an the 45-egree line. For given β, the equilibrium x an the critical payoff gain h = h β solve the equations f x ; h,β= 1, an 9 f x ; h,β= x 10 Using ientity 6 an then applying 10, Eq. 9 becomes βh + f x 1 = 1 f x Writing Eq. 10 explicitly an using 11 yiels: e β βh +x = x βh + x = 1 1 x e β 1 1 x = x The last equation uniquely ientifies x as a function of β. Eq. 11 can be rewritten as a quaratic equation: x x 1 = βh + There is a unique solution smaller than 1 given by x = βh + This equation implies that h is uniquely etermine given x, hence it is uniquely etermine given β. We now present an informal estimation of h β base on taing β to be very large. Claim establishes an upper boun for h β for all β>. For large β the equilibrium x tens to zero, hence the quaratic term in 1 is negligible. This means that x 1 βh 13 + Using this approximation twice, Eq. 10 becomes e β βh +x = x Rearranging terms yiels h β eβ β 1 x 1 + eβ e 1 β 1 βh + 14 This approximation is accurate for large β. However, for small β the term x in 1 is not negligible an the approximation breas own. Moreover, the expression in 14 is negative when evaluate at β = ; recall that h = 0. 1

11 G.E. Kreinler, H.P. Young / Games an Economic Behavior The next claim shows that if the expression in 14 is ajuste to equal zero at β =, it becomes a global upper boun for h β. Claim. Let β> an α hβ = eβ e β Then f x;α,β x > 0 on the interval [0, 1 ]. It follows that Γα,βexhibits fast selection. Note that hβ iffers from the expression in 14 only by a constant 4 e 1.8 in the numerator, an that h = h = 0. The proof of Claim is eferre to Appenix A. This conclues the proof that if β> an α hβ then Γα,β exhibits fast selection. We now show that when β 3 an α > hβ the process Γα,β exhibits fast selection to 99%. We claim that the high equilibrium is increasing in both α an β. Inee, ientity in footnote 13 implies that f / α is positive for all x > 0. We also have f β x; α,β= α + x 1 f x 1 f x By efinition x H > 1 f an thus β x H;α,β>0asclaime. It is thus sufficient to show that when β = 3 an α = h3 = e + 4 e/3 > 89%, the high equilibrium is above 99%. An explicit calculation shows that f 0.99; h3, 1 3 > > e It follows that x H > The final part of the proof is to show that the eterministic process is well approximate by the stochastic process for sufficiently large population size N. Let α an β be such that the process Γα,β isplays fast selection, namely there exists a unique equilibrium x H, an this equilibrium is strictly above 1. Given a small precision level δ>0, recall that T α,β,x H δ is the time until the eterministic process comes closer than δ to the equilibrium x H. Similarly, T N α,β,x H δ is the time until the stochastic process with population size N comes closer than δ to the equilibrium x H. Lemma. If the eterministic process Γα,β exhibits fast selection, then Γ N α,β also exhibits fast selection. More precisely, for any δ>0 we have lim ET Nα,β,x H δ = T α,β,x H δ 15 N Proof. The ey result is Lemma 1 in Benaïm an Weibull 003, which bouns the maximal eviation of the finite process from the eterministic approximation on a boune time interval, as the population size goes to infinity see also Kurtz, Before stating the result, we introuce some notation. Denote by x N τ the ranom variable escribing the aoption rate in the process Γ N α,β, where τ = 1 N an N. Toextentheprocessx N to a continuous-time process, efine the step process x N an the interpolate process ˆx N as x N t = x N τ, an ˆx N t = x N τ + t τ xn + 1τ xn τ τ for any t [τ, + 1τ. Lemma 3. Aapte from Lemma 1 in Benaïm an Weibull 003. For any T > 0 there exists a constant c = ct >0 such that for any μ > 0 an N sufficiently large: Pr xt ˆxN T μ e μ cn The proof is relegate to Appenix A. For convenience, we omit the epenence of T an T N on α an β. Assuming that T x H δ <, itisnoweasyto prove equality 15. Consier a small ɛ > 0, tae μ = ɛ an enote T ɛ = T x H δ + ɛ. Lemma 3 implies Pr x N T ɛ >x H δ + ɛ μ 1 e ɛ cn/4

12 50 G.E. Kreinler, H.P. Young / Games an Economic Behavior Fig. 5. Waiting times to within δ = 1% of the high equilibrium; 40 re, soli, 0 orange, ots an 10 green, ash ot revisions per capita. The x-axis is labele with the initial error rate ε; the corresponing noise level β appears in parentheses. It follows that T N Pr x H δ + ɛ T ɛ 1 e ɛ cn/4 We claim that PrT N < T q implies that ET N < T q.14 It follows that T ɛ ET N x H δ < ET N x H δ + ɛ/ 1 e ɛ cn/4 Taing limits in N on both sies we get that lim sup N ET N x H δ T ɛ for any ɛ > 0, hence lim sup N ET N x H δ T by taing ɛ 0. A similar argument shows that lim inf N ET N x H δ T. This conclues the proof of Theorem Theoretical bouns on waiting times We now embar on a systematic analysis of the magnitue of the time it taes to get close to the high equilibrium, starting from the all-b state. Fig. 5 shows simulations of the waiting times until the aoption rate is within δ = 1% of the high equilibrium. The blue, ashe line represents the critical payoff gain h β, such that for α > h β selection is fast. Pairs ε, α on the re, soli line correspon to waiting times T α,β,x H δ of 40 revisions per capita, while on the orange, otte an green, ash-otte lines the corresponing times are 0 an 10 revisions per capita, respectively. The following result provies a characterization of the expecte waiting time as a function of the payoff gain. Theorem. For any noise level β> an precision level δ>0 there exists a constant S = Sβ, δ such that for every α > h β an all sufficiently large N given α, β an δ, the expecte waiting time T N = T N α,β,x H δ until the aoption rate is within δ of the high equilibrium satisfies S ET N + log δ 1 + δ α+ h β The proof relies on the following simple argument. With probability q we have T N < T. With the remaining probability 1 q, fort T we now that xt is lower boune by a process y satisfying the same ifferential equation ẏ = f y y an with a ifferent starting point, namely yt = 0. This implies that with probability at least 1 qq we have T N < T. By iteration we obtain ET N q + 1 qq + 31 q q + T = q 1 q T = T q

13 G.E. Kreinler, H.P. Young / Games an Economic Behavior Fig. 6. An informal illustration of the evolution of the process. Fig. 7. Waiting time until the process is within δ = 1% of the high equilibrium, as a function of the payoff gain ifferential α h. Both axes have logarithmic scales. Simulation blue, ashe an upper boun re, soli. Initial error rates ε = 1% left panel an ε = 5% right panel. To unerstan Theorem, note that as the payoff gain α approaches the threshol h β, a bottlenec appears for intermeiate aoption rates, which slows own the process. Fig. 6 illustrates this phenomenon by highlighting the istance between the upating function f an the ientity function recall that the spee of the change of the process at aoption rate x is given by f x x. The first term on the right han sie of inequality 16 tens to infinity as α tens to h β, an the proof of Theorem shows that inequality 16 hols for the following explicit value of the constant S: S 0 = βh β+ When the payoff gain is large, the main constraining factor is the precision level δ, namely how close we want the process to approach the high equilibrium. The last two terms on the right han sie of inequality 16 tae care of this possibility. Fig. 7 plots, in log log format, the expecte time to get within δ = 1% of the high equilibrium as a function of α h,for ε = 1% left panel an ε = 5% right panel. These initial error rates correspon to β 4.59 an β.95 respectively. The constant S 0 taes the values S 0 1% 6 an S 0 5% 8. The re, soli line represents the estimate upper boun using constant S 0, while the blue, ashe line shows the simulate eterministic time T α,β,x H δ. Note that in both panels the two lines are parallel for small values of α h ; this shows that the rate of convergence of the waiting time presente in Theorem inverse square root is the correct rate. Finally, we give a concrete example of the estimate waiting time. As above, fix the initial error rate at ε = 5% an the precision level at δ = 1%. Then the critical payoff gain is h β 74%. Our estimate for the waiting time when α = 100% is T = 30.5, while the expecte waiting time for N = 1000 is T 1000 = Proof of Theorem. We shall show that Theorem hols for the eterministic approximation Γα,β. It then follows, using Lemma, that Theorem hols for all sufficiently large N given α,β an δ. The proof involves stuying the behavior of the waiting time T = T α,β,x H δ at the extremes, namely for α close to h = h β an for α large. The case of intermeiate α is covere by an appropriate choice of the constant S. Detaile proofs are relegate to Appenix B. 17

14 5 G.E. Kreinler, H.P. Young / Games an Economic Behavior Fig. 8. The function vx, α re, soli is a lower boun for f x; α green, ots. The essential ieas are as follows. When α approaches h from above, a bottlenec forms close to the position of the low equilibrium x L. The process slows own as it approaches the bottlenec, an the time it taes to traverse it increases as the bottlenec with gα ecreases. The term gα will be efine formally in the proof of the next lemma. The length of the bottlenec is controlle by the curvature of the response function at the tangential low equilibrium x L. The following lemma shows that as α approaches h the waiting time grows at a rate equal to the inverse square root of the with an length of the bottlenec. For convenience, we rop the epenence of f on β, which is hel fixe in the proof. Lemma 4. For any β>we have lim sup T α,β,x H δ f x α h L ; h gα 4 The iea of the proof of this lemma is epicte in Fig. 8. The waiting time is asymptotically the same when the response function f x;α,β is replace by the function vx;α,β, which is the upper envelope of the functions f x; h,β an x + gα. The bottlenec is the interval [x 1, x ] on which vx;α,β= x + gα. It can be shown that the time to approach the bottlenec, the time to traverse it, an the time to get away from it all grow at the same rate f x L ; h gα 1/. Our next result gives a linear estimate for the gap gα for α close to h. Lemma 5. lim inf α h h + α h f x L ; h 4 gα 1 βh + Combining the results in Lemmas 4 an 5 we conclue that the behavior of the waiting time as α approaches h is given by: lim sup α h T α,β,x H δ α + h βh + 18 Note that the right han sie of 18 is exactly S 0 as efine in 17. When α is large, the process slows own when it approaches the high equilibrium. The behavior in this case is controlle by the precision level δ, as escribe in the following lemma. Lemma 6. There exists α 0 such that for all α > α 0 T α; β,x H δ < log δ 1 + δ Theorem follows by putting together these results. Note that T α,β,x H δ is continuous in α, an Eq. 18 implies that there exists a constant S > 0 such that for all α h, α 0 ] S T α α+ h + 1 Together with Lemma 6, we fin that for any α > h

15 G.E. Kreinler, H.P. Young / Games an Economic Behavior S T α + log δ 1 + δ α+ h + 1 The exact result in Theorem follows by applying Lemma. 6. Partial information We now turn to the case where agents have a limite, finite capacity to gather information. Assume that each player samples other players before revising, where 3 is inepenent of N. As we shall see, partial information facilitates the sprea of the innovation, in the sense that the critical payoff gain is lower than in the full information case. Intuitively, for low aoption rates the effect of sample variability is asymmetric: the increase probability of aoption when aopters are over-represente in the sample outweighs the ecrease probability of aoption when aopters are uner-represente. In particular, the coarseness of a finite sample implies that the threshol h is no longer unboune as the noise tens to zero. 1 Inee, for α+ < 1,orequivalentlyα >, the existence of a single aopter in the sample maes it a best response to aopt the innovation. This implies that the process isplays fast selection for any noise level. This argument is formalize in Theorem 3 below. Here an for the remainer of the section, we moify the previous notation by aing as a parameter. For example, the process with payoff gain α, noise parameter β, population size N an sampling size is enote Γ N α,β,; the waiting time to aoption level p is enote T N α,β,, p an so forth. Theorem 3. Consier 3 <.Ifα > h β then Γ N α,β, isplays fast selection, where h β = min hβ, Proof. The proof follows the same logic as the proof of Theorem 1. Specifically, we shall show that the result hols for the eterministic approximation of the finite process, which implies that it also hols for sufficiently large population size. We begin by characterizing the response function in the partial information case. The next step is to show that, as in the case of full information, the high equilibrium of the eterministic process is unique. We then show that the threshol for partial information is below the threshol for full information, an also that forms an upper boun on the threshol for partial information. Detaile proofs are foun in Appenix C; here we shall outline the main steps in the argument. When agents have access to partial information only, the response function enote f x;α,β epens on the population aoption rate x as well as on the sample size. For notational convenience, fix the epenence of f an f on α an β an write f x an f x instea of f x;α,β an f x;α,β. The probability that exactly players in a ranomly selecte sample of size are playing A is x 1 x,forany = 0, 1,...,. In this case the agent chooses action A with probability f. Hence the agent chooses action A with probability f x = x 1 x f 19 =0 The efinition of the eterministic approximation of the stochastic process Γ N α,β, is analogous to the perfect information case. The continuous-time process Γα,β, has state variable xt that evolves accoring to the orinary ifferential equation ẋ = f x; α,β x, with x0 = 0 The stochastic process Γ N α,β, is well approximate by the process Γα,β, for large population size N. Inee,the statement an proof of Lemma apply without change to the partial information case. The following two lemmas state that the shape of the response function an the selection property establishe for full information continue to hol with partial information. Lemma 7. The function f is first strictly convex then strictly concave, an the inflection point is at most 1. Lemma 8. For any α > 0 there exists a unique equilibrium x H > 1 the high equilibrium, an it is stable. Furthermore, there exist at most two equilibria in the interval [0, 1 ]. The efinition of the critical payoff gain h β, is analogous to the efinition of h β in the proof of Theorem 1. With the convention inf =,let h β, = inf { α: Γα,β, isplays fast selection }