Local interactions and p-best response set

. We study a local interaction model where agents play a inite n -person game following a perturbed best-response process with inertia. We consider the concept of minimal p -best response set to analyze distributions of actions on the long run. We distinguish between two assumptions made by agents about the matching rule. We show that only actions contained in the minimal p - best response set can be selected provided that p is suiciently small. We demonstrate that these predictions are sensitive to the assumptions about the matching rule.


Introduction
here is an extensive literature on evolutionary game theory which investigates the long-run outcomes when a population of boundedly rational agents uses simple rules of adaptation to recurrently play a game in normal form.hese models allow for sharp-equilibrium selection results in some classes of games with multiple equilibria.Such an equilibrium can be interpreted as a convention, that is, a pattern of behavior that is customary and expected and regulates much of economic and social life (for a survey, see, among others, Young [1]).A common assumption in this literature is that interactions among agents are local and not global.Each individual may bematchedaccordingtosomematchingrulewithasubgroup of the overall population to play a game (for a survey, see, e.g., Weidenholzer [2]).Inthispaper,weassumethatagentshave no information about this matching rule.As a consequence, we elaborate two alternative scenarios corresponding to different assumptions made by agents about the matching rule.his point can be interpreted as a consequence of bounded rationality: agents only have an imperfect representation of their environment.We use the concept of (minimal) -best response set introduced by Tercieux [3]tomakepredictions for the long-run behavior of the population of agents (this concept is used by Durieu et al. [ ] in a ictitious play model with bounded memory).We establish that such predictions are possible for the whole class of inite -person games.We also study how assumptions about the matching rule afect these predictions.
Precisely, we consider a inite population of agents located at the vertices of a chordal ring, that is, a ring topology in which each vertex has additional links with other vertices.In particular, we focus on chordal rings of degree 2, ∈ N, constructed by adding at each vertex 2 − 2 chords with its closest vertices in a ring network.In order to deal with asymmetric inite -person games, we distinguish between classes of agents.A speciic role in the game is assigned to every class.Each agent of each class is located at exactly one vertex and two agents of the same class are located in diferent vertices.In this way, each vertex contains exactly agents.he game is played recurrently.At each period, one agent of each class is drawn at random to play the game.However, players' identity is not revealed.At the beginning of every period, each agent of each class observes the actions that the agents of other classes located in vertices linked with i t so w nv e r t e xh a sc h o s e ni nt h ep r e v i o u sp e r i o d .H eu s e s this information to estimate the probability distribution on the action proiles played by his potential opponents in the current period.hese estimations depend on the assumptions made by agents about the matching rule.Each agent believes that he can be matched with agents located at vertices linked with his own location.Precisely, we consider the following two scenarios.In the irst scenario, each agent assumes that his potential opponents are drawn jointly by location.In 2 Journal of Applied Mathematics the second scenario, each agent assumes that his potential opponents are drawn independently.For both scenarios, agents may have the opportunity to choose a best response to their estimation.he concept of -best response set allows us to study the way in which the distribution of actions that people take in the classes evolves over time.We show that, for both assumptions about the matching rule, only actions contained in the minimal -best response set can be selected on the long run provided that is suiciently small.For each assumption, an explicit bound of is given, and we analyze how this critical value evolves when increases.
Ellison [5,6] considers a similar local interaction structure: agents are arranged on a ring improved by adding chords between vertices in a regular form.However, since Ellison [5,6] focuses on symmetric two-player games, each vertex is associated with one agent and each agent has full information about the matching rule.Ellison [5] considers a symmetric 2×2coordination game. he model predicts that the riskdominant equilibrium is selected on the long run.Ellison [6] shows that in a symmetric ×game, the 1/2-dominant equilibrium, if it exists, is selected on the long run.Alós-Ferrer and Weidenholzer [7]considertheweakenedsolution concept of globally pairwise risk-dominant equilibrium and show that this equilibrium, if it exists, is selected in a symmetric 3×3coordination game played by agents arranged on a ring.However, a generalization of this result to a larger class of games is not possible.Indeed, Alós-Ferrer and Weidenholzer [7] present an example of a 4×4coordination game in which the globally pairwise risk-dominant equilibrium is n o ts e l e c t e do nt h el o n gr u n .F r o mt h i sp o i n to fv i e w ,t h e conceptofminimal-best response set is interesting.Since a minimal -best response exists in every game, it allows us to investigate the long-run outcomes in the whole class of inite -person games.Moreover, since the concept of minimal -best response set generalizes the concept of -dominant equilibrium [8], the result of Ellison [6]i sap a rt i cu l a rc a se of our result.h ep a p e ri ss t r u c t u r e da sf o l l o w s .Section 2 introduces notations.Section 3 introduces the concept of -best response set.Section introduces the learning model.In Section 5 we present the main results.Section 6 concludes.

Notations and Definitions
Let ⊆ denote weak set inclusion and let ⊂ denote proper set inclusion.We denote by ⌈⌉ the smallest integer greater than or equal to .Foranyiniteset, Δ() denotes the set of all probability distributions on .
Let Γ be a inite -person strategic-form game.Let b et h ei n i t es e to fp u r es t r a t e g i e s available to player ∈ = {1,2,...,}.W ew r i t eΔ( ) f o rt h es e to fp r o b a b i l i t y distributions over for each ∈ .L e t ( ) denote the probability mass on strategy .D e i n et h ep r o d u c ts e t =∏ ∈ .LetΔ() be the set of probability distributions on .L e t − =∏ ̸ = denote the set of all possible combinations of strategies for the players other than ,wi th generic elements − =( ) ̸ = .L e tΔ( − ) be the set of probability distributions on − with generic elements − .We sometimes identify the element of Δ( ) that assigns probability one to a strategy in with this strategy in .
In this paper, a player's belief about others' strategies takes the form of a probability measure on the product of all opponents' strategy sets.We assume that each player has expected payofs represented by the function : × Δ( − )→R.
For each player and probability distribution − ∈ Δ( − ),let be the set of pure best responses of against − .Let =∏ ∈ be a product set where each is a nonempty subset of .Le tΔ( − ) denote the set of probability distributions with support in − .F i n a l l y , (Δ( − )) denotes the set of strategies in that are pure best responses of against some distribution − with support in − ;thatis, (2)

-Best Response Sets
We will now introduce the concept of (minimal) -best response set.Let ∈[ 0 , 1 ] and let − be a nonempty subset of .L e t (Δ( − ,)) denote the set of strategies in that are pure best responses by to some distribution − ∈ Δ( − ,)(regardless of probability assigned to other possible combinations of strategies); that is, Let us recall the deinition of a strict -dominant equilibrium irst introduced by Morris et al. [8]intwo-persongamesand extended to -person games by Kajii and Morris [9].A proile In the sequel, we focus on the case where = for all ∈ .he concept of -best response set extends the concept of strict -dominant equilibrium to product sets of strategies.Formally, a (minimal) -best response set is deined as follows.
Deinition 1.Let ∈[0,1].A-best response set is a product set ⊆, where for each player A -best response set is a minimal -best response set if no -best response set is a proper subset of .
Let Q be the collection of -best response sets for some ∈[ 0 , 1 ] .he following lemma states some properties of minimal -best response sets.Lemma 2. Let Γ be a inite -person game.
For a proof of this Lemma, we refer the reader to Durieu et al. [ ]andT ercieux [3].

Adaptive Processes
Following Samuelson [10], we extend the Darwinian process proposed by Kandori et al. [11]tothem ul ti popula tio ncase.his extension is quite natural to deal with asymmetricperson games.We think the game Γ as having roles.For each role ∈ , a nonempty class ofindividualsiseligible to play that role.We assume that each class is composed of identical agents, where is a inite integer.
We consider the possibility of local interactions.Let (, ) be a graph where ={ 1 , 2 ,..., } is the set of vertices and is the set of edges.We assume that vertices are located increasingly in a clockwise direction around a ring.We fo c us on a chorda l r ing: ver tex , ∈ {1,2,...,} is adjacent to the vertices (modulo ) ±1 , ±2 ,..., ± where 2 ≤ − 1.If2 = − 1, then each vertex is adjacent to each other vertex.For each vertex ∈ , denote by () the open neighborhood of ,t h a ti s ,t h es e to fv e r t i c e sa d j a c e n t to .S i n c e|()| = 2 for each ∈ , (, ) is regular.he closed neighborhood of , denoted by (), is deined by () ∪ {}.
We assume that each class of agents is distributed among the set of vertices .In other words, each agent of each class , ∈ , is located at exactly one vertex of and two agentsofthesameclassarelocatedindiferentverticesof.In this way, each vertex contains exactly agents.From this point of view, each vertex can be interpreted as a location.We denote by () ∈ the actions proile chosen by agents located in ∈. he collection of action proiles chosen in all locations in is Θ= .
Let = 1, 2, . . .denote successive time periods.he stage game Γ is played once in each period.In period ,one agent is drawn at random from each of the classes and assigned to play the corresponding role.We assume that, at every period, each agent has no information about who is selected to play each role in the game, only that a given action is played by someone.his lack of information gives rise to two kinds of assumptions made by each agent about his potential opponents, that is, about the matching rule.Fix a location ∈ .In a irst scenario, each agent in ∈ assumes that his potential opponents belong to a unique location in ().In other words, opponents would be drawn jointly by location in the closed neighborhood of his location.
In a second scenario, each agent in ∈assumes that his potential opponents may belong to diferent locations in ().
In other words, opponents would be drawn independently in the closed neighborhood of his location.We give a formal description of both scenarios.Consider a time period .Actions chosen in by the whole population are described by =( ( 1 ), ( 2 ),..., ( )) ∈ Θ.Fixan agent ∈ located in and an action proile ∈ .Denote by ( − ) the number of locations in () such that − () = − in .According to the irst scenario, at the beginning of period +1,agent believes that the probability to be matched with agents playing − is In order to describe the second scenario, it is convenient for each class to identify with the set {1,2,..., },where =| |.F o re a c h∈ such that ̸ = ,a n de a c h ∈ {1,..., }, denote by ( ) the number of () where is chosen in .Consider − =( ) ̸ = ∈ − .According t ot h es e c o n ds c e n a r i o ,a tt h eb e g i n n i n go fp e r i o d+1 , agent believes that the probability to be matched with agents playing − is In period ,everyagentineachclass chooses an action ∈ according to the following learning rule.Every agent might receive the opportunity to revise his choice.For the sake of simplicity, we assume that this adjustment probability does not depend on the agent nor on the actions chosen in the whole population.Whenever an agent does not receive a revision opportunity, he simply repeats the action he has taken in the past.Whenever an agent receives a revision opportunity, he switches to a myopic best response.h a ti s ,t h ea g e n ta s s u m e st h a tt h ea c t i o nc h o i c e so fo t h e r agents will remain unchanged in the next period and adopts ap u r eb e s tr e s p o n s ea g a i n s ti t .P r e c i s e l y ,a g e n t chooses a pure best response against the probability distribution on − computed using formula ( 6)or (7).Inotherwords,wedeine two myopic best-response dynamics with inertia.
If probability distributions are computed using (6), the d y n a m i c sr e s u l t si naM a r k o vc h a i no nt h es t a t es p a c eΘ, denoted by 0  .We refer to it as best-response process with joint drawing by location.Similarly,ifprobabilitydistributions are computed using (7), the dynamics results in a Markov c ha ino nth es ta t es paceΘ, denoted by 0  .We refer to it as best-response process with independent drawing.LetR (resp.R ) be the collection of recurrent sets of 0 (resp., 0 ).For each ∈and each subset Θ ⊂Θ ,l e t (Θ ) be the set of actions in that appears in Θ .hep r o d u c ts e to fa l l actions that appears in Θ is (Θ )=∏ ∈ (Θ).Consider a minimal -best response set .DenotebyR ⊆ R (resp., R ⊆ R ) the collection of recurrent sets of 0 (resp. 0 response set and thus a 1-best response set.

Journal of Applied Mathematics
Following the literature (see, e.g., [11,12]), the model is completed by adding the possibility of rare mutations or experiments on the part of agents.With ixed probability >0, independent across players and across time, each agent chooses an action at random. he processes with mutations are called perturbed processes and are denoted by and .With these mutations as part of the processes, each state of Θ is reachable with positive probability from every other state.Hence, the perturbed processes and are irreducible and aperiodic inite state Markov chains on Θ.Consequently ,for each >0, (resp., ) has a unique stationary distribution (resp., )s a t i s fy i n g = (resp., = ).he limit stationary distribution (as the rate of mutation tends to zero) * = lim →0 (resp., * = lim →0 ) exists and is a stationary distribution of the unperturbed process 0 (resp., 0 ). he states in the support of * (resp.* )a r e called stochastically stable states and form a subset of R (resp., R ). he recurrent sets appearing in the support of * (or * )a r eth o sewh i c ha r eth eea s i e s tt or ea c hfr o mal l other recurrent sets, with "easiest" interpreted as requiring the fewest mutations (cf., heorem in [12]).
W er e l yo nt h ei d e n t i i c a t i o no ft h es e to fs t o c h a s t i c a l l y stable states developed by Ellison [6].his identiication proceeds as follows.Ellison [6] introduces suicient conditions to have a collection of recurrent sets that contains all stochastically stable states.he analysis uses three measures: t h er a d i u s ,t h ec o r a d i u s ,a n dt h em o d i i e dc o r a d i u s .T o illustrate these concepts, consider a collection of recurrent sets Θ 1 ⊆ R .L e t(Θ 1 ) be the basin of attraction of Θ 1 , that is, the set of states ∈Θfrom which the unperturbed process converges to Θ 1 with probability one.In other words, (Θ 1 ) is the set of states ∈Θ such that it is possible without mutation to build a path, that is, a sequence of distinct states, from to 1 ∈Θ 1 .her a d i u so f( t h eb a s i n of attraction of) Θ 1 , denoted by (Θ 1 ),i st h em i n i m u m number of mutations necessary to leave (Θ 1 ) from Θ 1 ,that is, the minimum number of mutations contained in any path from a state 1 ∈Θ 1 to a state 2 ∉ ( Θ 1 ). he coradius of (a basin of attraction of) Θ 1 , denoted by (Θ 1 ),i st h e maximum over all states 2 ∉Θ 1 of the minimum number of mutations necessary to reach (Θ 1 ),thatis,themaximum over all states 2 ∉Θ 1 of the minimum number of mutations contained in any path from 2 to a state 1 ∈ ( Θ 1 ).T o c o m p u t et h em o d i i e dc o r a d i u so f( t h eb a s i no fa t t r a c t i o n of) Θ 1 , consider a state 2 ∉Θ 1 and a path from 2 to a state belonging to Θ 1 .he modiied number of mutations of this path is obtained by subtracting from the number of mutations of the path the radius of the intermediate recurrent sets through which the path passes.he modiied coradius of (a basin of attraction of) Θ 1 , denoted by * (Θ 1 ),i s the maximum over all states 2 ∉Θ 1 of the minimum modiiednumberofmutationsnecessarytoreach(Θ 1 ),that is, the maximum over all states 2 ∉Θ 1 of the minimum modiied number of mutations associated with any path from 2 to a state 1 ∈ ( Θ 1 ).N o t et h a t * (Θ 1 )≤ ( Θ 1 ) for every Θ 1 ⊆Θ .E l l i s o n [ 6]e s t a b l i s h e st h ef o l l o w i n g result.
Since * (Θ 1 ) ≤ (Θ 1 ),a na l t e r n a t i v ec o n d i t i o nt o have all stochastically stable states contained in Θ 1 is that (Θ 1 ) > (Θ 1 ).

Selection Results
Fix a minimal -best response set .First, consider the perturbed best-response process with joint drawing by location.We give a suicient condition to have all stochastically stable states associated with .heorem 4. Let Γ be a inite -person game and (, ) a c h o r d a lr i n go fd e g r e e2.L e t≤1 / and let be the minimal -best response set of Γ.If is suiciently large and is suiciently small, the perturbed process puts arbitrarily high probability on a subset Θ * ⊆Θsuch that (Θ * )⊆.
Proof.Observe that if R = R , then the result follows.In the sequel, we assume that R ⊂ R .Webreaktheproofinto three parts.
(1) We g ive a lower b ound on (R ).Fixastate∈R , alocation∈, and two distinct classes and .Consider agent ∈ located in and agent ∈ located in ∈ ().Assume that agent mutates: he chooses at random a strategy not contained in .hen,agentbelieves that the probability to be matched with agents playing a vector of actions not contained in − is ∑ − ∉ − − ( − ) = 1 / ( 2 +1 ) .N o w , assume that ≤2 +1agents in located in locations belonging to () mutate and choose an action outside .hen, agent believes that the probability to be matched with agents playing a vector of actions not contained in − is ∑ − ∉ − − ( − )=/(2+1).Let be a probability such that at least one class of agents has an action ∉ as a pure best response to − ,where∑ − ∈ − − ( − )=.By deinition of a minimal -best response set, we have >.A transition from Θ 1 ∈ R to any Θ 2 ∉ R requires at least mutations (in locations belonging to () and outside ), where is such that /(2 + 1) ≥ 1 − .O t h e r w i s e ,a g e n t's b estresponse(s) to − belong(s) to since We g ive an upp er b ound on * (R ).Todothis,itis convenient to distinguish between two situations according to the values of and .Firstly, consider the cases such that ⌈(2 + 1)⌉ ≤ .F i xas t a t e∉R and a class .Assume that, in a location ∈ , ( − 1) agents ∉ mutateandchooseactionsin .hen,agent∈located in ∈( ) believes that the probability to be matched with agents playing a vector of actions contained in − is ∑ − ∈ − − ( − ) = 1/(2 + 1).Now, consider ≤2 +1 consecutive locations in (, ) denoted by , ∈ {1,...,}.Assume that in each of these locations (−1)agents ∉ mutateandchooseactionsin .hen,agent∈located the probability to be matched with agents playing a vector of actions contained in − is ∑ − ∈ − − ( − ) = (1/(2 + 1)) −1 .Now, consider ≤2 +1consecutive locations in (, ) denoted by , ∈ {1,...,}.Assume that in each of these locations ( − 1) agents ∉ m u t a t ea n dc h o o s ea c t i o n s in .hen,agent∈located in such that ∈( ) for each ∈ {1,...,}believes that the probability to be matched with agents playing a vector of actions contained in − is By deinition of a minimal -best response set, agent has at least one strategy ∈ as a pure best response to − ater that all agents ∉ located in * ≤2 +1consecutive locations in (, ) belonging to () m u t a t ea n dc h o o s ea c t i o n si n if * is such that ( * / ( 2 +1 ) ) −1 ≥ .S e t * =⌈ 1/(−1) (2 + 1)⌉.S i n c e ≤ 1− ,ifin * consecutive locations every agent ∉ mutates and chooses an action in ,theneveryagent∈ located in each of these locations chooses an action in .By inertia, it is then possible to reach a state in which an action proile contained in is played in * consecutive locations in (, ).From such a state, it is possible to reach a state in R without additional mutation.Indeed, since ≤ 1− and >2 ,wenecessarilyhave * ≤ .Hence,(R A suicient condition to obtain inequality (11)isthat Inequality ( 12) is satisied provided is suiciently large since by hypothesis ≤ 1− ,and,bydeinitionofaminimal-best response set, we have 1−>1−.
he following result is an immediate application of heorem 6.

Corollary 7.
Let Γ be a inite -person game and (, ) ac h o r d a lr i n go fd e g r e e2.L e t * be a strict -dominant equilibrium of Γ,where≤ −1 .If is suiciently large and is suiciently small, the perturbed process puts arbitrarily high probability on a subset Θ * ⊆Θsuch that (Θ * )={ * }.

Conclusion
his paper establishes that the concept of minimal -best response set is useful to study the long-run outcomes of a p r o c e s sw h e na g e n t sa r ea r r a n g e do nac h o r d a lr i n ga n d follow a myopic best response rule with inertia.In particular, it allows us to obtain results for the whole class of initeperson games.Even if predictions are not necessarily sharp (since a minimal -best response set may become large when decreases), those results make easier the identiication of stochastically stable states.he paper also highlights that predictions depend on the assumptions made by agents about the matching rule.From this point of view, it is possible to establish a connection between these results and the results obtained in Durieu et al. [ ]. his paper considers a ictitious play model with bounded memory and sample as in Young's [12].Two processes are studied.On the one hand, it is assumed that each agent believes that, in every period, his opponents play independently.On the other hand, each agent believes that, in every period, the play of his opponents is correlated.Durieu et al. show that the concept of -best response set allows establishing predictions about the longrun outcomes of both processes.Furthermore, as in the present paper, there exists a similar gap between predictions obtained for each process.his conveys the idea that sampling in memory and believing that opponents correlate their actions (play independently resp.) has the same efect as believing that players are drawn jointly (independently resp.) in neighborhood to play the game.