1. Introduction. 2. A Simple Model

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2 . Introduction In the last years, evolutionary-game theory has provided robust solutions to the problem of selection among multiple Nash equilibria in symmetric coordination games (Samuelson, 997). More specically, a series of papers has suggested that, whenever the game is played by a population of myopic agents who are repeatedly matched over time and employ bestreply rules to update their strategies, the risk-dominant (RD) Nash equilibrium will be almost always preferred to the Pareto-efficient (PE) one in the long-run. These models have been often employed to address the issue of technological adoption in the presence of several competing standards and network externality effects (cf. e.g., Kirman, 997). However, in such formalizations, agents typically belong to a fixed-size population and are always able to revise the strategy they currently play. This might seem to be at odds with many real-world adoption choices, where agents sequentially enter the market and irreversibly choose one of the existing standards (Katz and Shapiro, 986; Nelson, Peterhansl, and Sampat, 004). On the contrary, irreversible adoption with sequential entry has been studied in Polya- Urn models (Arthur et al., 987; Dosi and Kaniovski, 994). Yet, these models did not explicitly address the issue of selection among equilibria (e.g., adoption shares) that can be ranked according to risk-efficiency criteria. To fill this gap, this note begins to explore equilibrium selection in symmetric coordination games played by myopic agents where: (i) the cardinality of the population increases over time due to a positive net entry rate; (ii) players irreversibly choose a strategy when they enter the economy. We employ a standard Polya-Urn framework to describe the dynamics of strategy shares. More precisely, we assume that one agent enters the economy in any time period t =,,... and must irreversibly choose among the available strategies with probabilities which depend on expected payoffs. We study the long-run behavior of the system under alternative individual decision rules (e.g. best-replies with or without stochastic mistakes, linear or logit probabilistic rules) and we characterize its efficiency properties. The rest of this note is organized as follows. Section describes the model. In Section 3 we investigate the long-run behavior of the system and we discuss equilibrium selection issues. Section 4 concludes.. A Simple Model Consider an economy evolving in discrete time-steps t = 0,,,... At time t = 0, there are N 0 > 0 agents in the economy. Each agent is fully described by a binary pure strategy (strategy, henceforth) s S = {, +}. Thus, at any t, the state of the system can be characterized by the current share x t [0, ] of agents playing +. Cf. e.g. Kandori et al. (993) and Young (996). This conclusion holds true also agents always play the game against their nearest-neighbors (Ellison, 993) and, to some extent, they are able to endogenously choose their opponents in the game (cf. Jackson and Watts, 00; Goyal and Vega-Redondo, 003; Fagiolo, 005).

3 The dynamics runs as follows. Given some initial share x 0, at any t > 0 one new agent enters the economy and irreversibly chooses a strategy in S. This implies that the size of the population at time t is N t = N 0 + t. The entrant then plays a symmetric coordination game against all incumbent agents according to a standard stage-game payoff matrix as in Table, left panel. + + a b c d α β Table : Stage-Game Payoff Matrix. Left: Original Payoffs. Right: Normalized Payoffs. In the left panel of Table, a > c and b < d because the game is a coordination one. We also assume that a > d (i.e. the Nash equilibrium (+, +) is Pareto efficient by construction) and that a b. In what follows, however, we shall focus on the -parameter stage-game payoff matrix in the right panel of Table, obtained from the former without loosing in generality by letting α = (c b)/(a b) and β = (d b)/(a b). Given the above restrictions, α < and 0 < β. For the sake of convenience, we also assume that α 0. Notice that β = and α = 0 the game is a pure-coordination one. Moreover, α + β = the two Nash equilibria E + = (+, +) and E = (, ) are risk-equivalent. Thus, α + β <, E + is both PE and RD, while, α + β >, E + is PE but E is RD. Since agents are myopic, expected payoffs associated to any given choice s S are given by: { π t (s) = π(s, x t ; α, β) = x t (α β)x t + β s = + s =. () Let us call Pr{s ; π + t, π t } the probability that the agent chooses s S. We assume that entrants choose s S according to one of the individual decision rules in Table. The dynamics can be easily formalized in terms of a Polya-Urn framework (Arthur et al., 984; Dosi et al., 994). Since, under any decision rule, Pr{s ; π + t, π t } only depends through payoffs on x t, α and β, we can define: Thus, the dynamics of x t reads: p(x t ; α, β) = Pr{s ; π + t, π t }. () x t+ = x t + p(x t; α, β) x t N 0 + t + + η(x t; α, β) N 0 + t +, (3) where η t is a zero-mean, dichotomous random variable such that: Pr{η(x t ; α, β) = η} = { p(xt ; α, β) p(x t ; α, β) η = p(x t ; α, β) η = p(x t ; α, β). (4) Best-reply rules are studied in Blume (995). Linear-probability and logit rules are discussed in Anderson, depalme, and Thisse (99), Blume (993), Weisbuch, Kirman, and Herreiner (000) and Brock and Durlauf (00).

4 Acronym Decision Rule Pr{s ; π + t, π t } DBR PBR Deterministic Best-Reply Perturbed Best-Reply / 0 ɛ / ɛ π t (s ) > π t ( s ) π t (s ) = π t ( s ) π t (s ) < π t ( s ) π t (s ) > π t ( s ) π t (s ) = π t ( s ) π t (s ) < π t ( s ) LP NP Linear Probability Non-Linear (logit) Probability π t (s ) π t (+)+π t ( ) exp{γ π t (s )} exp{γ π t (+)}+exp{γ π t ( )} Table : Individual Decision Rules. PBR: ɛ > 0. NP: γ > 0. Note: (i) PBR converges to a DBR as ɛ 0; (ii) NP converges to a DBR as γ. As t increases, the impact of stochastic entry choices on x t becomes weaker and weaker. Dosi et al. (994) show that, under some mild restrictions, x t converges with probability to a stable point belonging to the zeroes of the function g(x) = p(x) x, where x [0, ] 3. The properties of p determine the conceivable limit points of the system. However, the dynamics is path-dependent: there are at least two stable points, which one is actually selected by the process depends on both initial conditions (x 0 ) and on the stochastic sequence of early entrants decisions. 3. Long-Run Behavior and Equilibrium Selection In this Section, we characterize the limit properties of the system under the four decision rules in Table. Let us start with a DBR rule. In this case, it is easy to see that: Proposition If agents choose under a DBR rule, then: x t > x p(x t ; α, β) = x t = x, 0 x t < x where x = β/( α + β). Thus: Pr{lim t x t {0, } } =. Proof. See Appendix A. Thus, full coordination will always emerge in the limit, but we cannot predict it will occur on E + or E. Notice, however, that the larger x, the smaller the set {x [0, ] : 3 See Appendix A. In general, a root x (0, ) of g(x) is stable g(x) crosses the line y = x from above at x. A similar definition applies for x {0, }. 3

5 p(x) > x}. Since x increases with β/( α), the more E is risk-dominant (i.e. the more β > α), the larger is the set of shares for which, on average, x t+ decreases to 0. If the two equilibria are risk-equivalent (α + β = ), then x =. So, one supposes to draw x 0 from a random variable unormly distributed over [0, ], the value of x can give us a measure of the likelihood of the RD equilibrium in the deterministic dynamics: E(x t+ x t ) = x t + p(x t; α, β) x t. (5) N 0 + t + When agents instead employ a PBR, the probability with which the currently beststrategy is chosen becomes ɛ, while the currently inferior option is now selected with probability ɛ. We have that: Proposition If agents choose under a PBR rule, then: ɛ x t > x p(x t ; α, β) = x t = x, ɛ x t < x where x = β/( α + β). Thus Pr{lim t x t {ɛ, ɛ}} =. Proof. See Appendix A. Coexistence of both + and players will always emerge in equilibrium. If the system tends to select the PE (respectively, the RD) equilibrium, then an ɛ-sized niche of agents playing the RD, inefficient equilibrium (respectively, the PE one) will always survive. As happens with the DBR rule, the long-run coordination outcome is entirely unpredictable. However, the larger x (i.e. the stronger risk-dominance of E ), the larger the probability that one draws initial conditions at random the deterministic dynamics in (5) will select the RD outcome. The parameter ɛ is typically interpreted as the probability that an agent makes a mistake with respect to the optimal available decision. This mistake occurs independently of how large (relative) payoffs are. More generally, one might suppose that agents choose only in probability the strategy associated to the highest payoffs (LP or NP). If agents employ a LPR, one can prove: Proposition 3 If agents choose under a LP rule, then: p(x t ; α, β) = where α [0, ) and β (0, ]. Hence: x t ( + α β)x t + β,. If α = 0 and β =, then x [0, ], Pr{lim t x t = x} > 0.. If α 0, let x = β β+α. Then: Pr{lim t x t = x } =. 4

6 Proof. See Appendix A. Under a LP rule, and the game is not a pure-coordination one, the long-run behavior of the system becomes predictable: the share of agents playing + in the limit will converge a.s. to x. Notice that x > 0 unless β = : full coordination on the RD outcome will never be the case unless E + and E are Pareto-equivalent. Full coordination on the PE outcome will conversely emerge only α = 0 (that is, the loss from miscoordination is the same for both strategies). But what happens in terms of PE vs. RD equilibria? It is easy to see that x > α + β <. If the two equilibria are risk-equivalent then x =. Conversely, the larger α+β >, the smaller the fraction of agents playing + in equilibrium. However, there will always exist in the limit a small niche of players choosing the superior strategy. Similarly, E + is both PE and RD and α + β < approaches 0, then the system will converge to a x > which grows with α + β. Unless α = 0, there will always be a persistent niche of players in equilibrium. Suppose now that entrants choose their strategy according to a non-linear, logit probability rule. Then: Proposition 4 If agents choose under a NP rule, then: p(x t ; α, β, γ) = α γβ[ + e β xt +xt] = + e θ[σxt +xt] = p(x t ; θ, σ), (6) where θ = γβ > 0 and σ = α β > 0. Therefore:. If α + β = σ = then: If θ then Pr{lim t x t = } =. If θ > then {x(θ), x(θ)} : Pr{lim t x t {x(θ), x(θ)}} =, where: x(θ) <, x(θ) >, and, as θ, x(θ) 0 and x(θ).. If α + β σ then numerical analyses show that: If θ then x(θ, σ) : Pr{lim t x t = x(θ, σ)} =. If σ < α + β > then x(θ, σ) < and decreases with θ. If σ > α + β < then x(θ, σ) > and x(θ, σ) and increases with θ. Moreover, x(θ, σ) as σ for any θ. If θ > then (3) admits both stable root and stable roots. However, σ > α + β < and θ is sufficiently large, then stable roots {x(θ, σ), x(θ, σ)} always emerge. As before, we have that x(θ, σ) <, x(θ, σ) > and, as θ, x(θ, σ) 0, x(θ, σ). Proof. See Appendix A. Under a NP rule, the long-run behavior of the system is predictable θ. This happens either when the payoff of (, ) is small (β small), or when entrants choices are not very sensitive to current payoffs (γ small). Thus, risk-dominance of E + is not sufficient to guarantee predictability. However, when θ coexistence always occurs and system 5

7 efficiency as measured by x(θ, σ) increases the more E + is RD. On the contrary, θ > and (E +, E ) are risk-equivalent, two stable roots always arise and they tend to diverge toward 0 and as θ increases. This regime also occurs when agents become very sensible to current payoffs and E + is RD. In such a case, coexistence on efficient (x(θ, σ) > ) or inefficient (x(θ, σ) < ) mixes can always appear. 4. Concluding Remarks In this note, we studied equilibrium selection in a symmetric coordination game played by myopic agents in a population whose size increases over time. We have shown that the system delivers a predictable outcome only agents employ either a linear probability rule or they use a logit rule, but the latter is not very sensible to current payoffs. Full coordination on the PE outcome can only occur agents employ: (i) deterministic best-reply rules (but the actual outcome depends here on initial conditions and chance); or (ii) linear probability rules and the loss from miscoordination is the same for both strategies. In all other cases, coexistence of strategies will emerge in the long run and the efficiency level of the equilibrium configuration will greatly vary with initial conditions, early entrants choices and stage-game payoffs. References Anderson, S., A. depalme and J.-F. Thisse (99), Discrete Choice Theory of Product Dferentiation. Cambridge, MIT Press. Arthur, W., Y. Ermoliev and Y. Kaniovski (984), Strong laws for a class of path-dependent stochastic processes with applications, in Argin, S. and P. Shiryayev (eds.), Proceedings Conference on Stochastic Optimization. Lecture Notes in Control and Information Sciences. Berlin, Springer Verlag. Arthur, W., Y. Ermoliev and Y. Kaniovski (987), Path-dependent processes and the emergence of macro-structure, European Journal of Operational Research, 30(): Blume, L. (993), The statistical mechanics of strategic interaction, Games and Economic Behaviour, 5: Blume, L. (995), The statistical mechanics of best-response strategy revision, Games and Economic Behavior, : 45. Brock, W. and S. Durlauf (00), Discrete Choice with Social Interactions, Review of Economic Studies, 68: Dosi, G., Y. Ermoliev and M. Kaniovski, Yu. (994), Generalized urn schemes and technological dynamics, Journal of Mathematical Economics, 3: 9. Dosi, G. and Y. Kaniovski (994), On badly behaved dynamics, Journal of Evolutionary Economics, 4:

8 Ellison, G. (993), Learning, local interaction and coordination, Econometrica, 6: Fagiolo, G. (005), Endogenous Neighborhood Formation in a Local Coordination Model with Negative Network Externalities, Journal of Economic Dynamics and Control, 9: Goyal, S. and F. Vega-Redondo (003), Network Formation and Social Coordination, Working Paper No.48, Department of Economics, Queen Mary, University of London. Jackson, M. and A. Watts (00), On the formation of interaction networks in social coordination games, Games and Economic Behavior, 4: Kandori, M., G. Mailath and R. Rob (993), Learning, mutation and long run equilibria in games, Econometrica, 6: Katz, M. and C. Shapiro (986), Technology Adoption in the Presence of Network Externalities, Journal of Political Economy, 94: Kirman, A. (997), The Economy as an Evolving Network, Journal of Evolutionary Economics, 7: Nelson, R., A. Peterhansl and B. Sampat (004), Why and how innovations get adopted: a tale of four models, Industrial and Corporate Change, 3: Samuelson, L. (997), Evolutionary Games and Equilibrium Selection. Cambridge, The MIT Press. Weisbuch, G., A. Kirman and D. Herreiner (000), Market organisation and trading relationships, Economic Journal, 0: Young, H. (996), The Economics of Convention, Journal of Economic Perspectives, 0: 05. 7

9 Appendix: Proofs To prove Propositions -4 above, we will make use of the following theorems from Dosi, Ermoliev, and Kaniovski (994). Let x t the proportion of + players at time t = 0,,,... and define the set of zeroes of the function g(x) = p(x) x as: B = {x [0, ] : [ψ(x), ψ(x)] 0}, where ψ(x) = lim inf y x [p(y) y], ψ(x) = lim sup y x [p(y) y] and y belongs to R(0, ), i.e. the set of rational numbers in (0, ). Then: Theorem (Arthur et al., 984). The sequence {x t } converges a.s. to the set B as t. An isolated point ξ B is called stable for every (ε, ε ) R ++ small enough then g(x)(x ξ) < (ε, ε ) < 0, provided that ε x ξ ε and x belongs to R(0, ) Conversely, an isolated point ξ B is called unstable for every ε > 0 small enough, then g(x)(x ξ) > 0 for any x R(0, ) [(ξ ε, ξ) (ξ, ξ + ε)]. Thus: Theorem (Dosi et al., 994).. If ξ B is a stable point in (0, ) and there exists (ε, ε ) R ++ such that p(x) > 0 for x R(0, ) (ξ ε, ξ) and p(x) < for x R(0, ) (ξ, ξ +ε ), then Pr{lim t x t = ξ } > 0 for every x 0 (ξ ε, ξ + ε ).. If ξ B is an unstable point in (0, ) and one of the following conditions hold: (a) in a neighborhood of ξ the function p is continuous; (b) there exists ε > 0 s.t. for x R(0, ) (ξ ε, ξ + ε) then g(x)(x ξ) λ x ξ +η and p(x)( p(x)) η > 0, where η > η 3 (λ + ), η [0, ], λ > 0 and η 3 = sup x R(0,) (ξ ε,ξ+ε) p(x)[ p(x)]; (c) there exists a neighborhood of ξ where for x ξ (resp. x ξ) either (a) or (b) holds true and for x > ξ (resp. x < ξ) p(x) = (resp. p(x) = 0). Then: Pr{lim t x t = ξ} = 0 for every x Let y 0 t = N o x o /N t and y t = +y 0 t N o /N t. If p(y 0 t ) < for t 0 and t 0 p(y0 t ) < then Pr{lim t x t = 0 } > 0. Also p(y t ) > 0 for t 0 and t 0 [ p(y t )] < then Pr{lim t x t = } > 0. We are now well-equipped to prove Propositions -4 above. Proof of Propositions - To obtain p(x t ; α, β) in both cases, it suffices to note that π t (+) π t ( ) and only x t > x = β/( α + β). Define for ɛ 0 a generic BR rule as: ɛ x t > x p(x t ; α, β) = x t = x. ɛ x t < x 8

10 Of course, ɛ = 0 we have a DBR, while a PBR is recovered for ɛ > 0. It is easy to see that B = {ɛ, x, ɛ}. By applying Theorem, points (-3), above one has that the set of stable points is B (ɛ) = {ɛ, ɛ}. Therefore, Pr{lim t x t B (ɛ)} =. This means that: (i) under a DBR, Pr{lim t x t {0, }} = ; and (ii) under a PBR, Pr{lim t x t {ɛ, ɛ}} =. Figures and depict two examples of p(x t ; α, β) under DBR and PBR rules x * p(x) x Figure : An example of p(x t ; α, β) with DBR Rules (α = 0.3, β = 0.5, x = /3) x * p(x) x Figure : An example of p(x t ; α, β) with PBR rules (ɛ = 0., α = 0.3, β = 0.5, x = /3). 9

11 By replacing expected payoffs () in: one gets Proof of Proposition 3 p(x t ; α, β) = p(x t ; α, β) = π t (+) π t (+) + π t ( ), x t ( + α β)x t + β. Consider first the case α = 0 and β =. Here, p(x t ; α, β) = x t. Therefore, B = [0, ] and Pr{lim t x t = x} > 0 for all x B. If (α, β) (0, ), it is easy to see that p is C 0 over [0, ] and sign( p/ x t ) = sign(β) > 0 x t [0, ]. Since p(0; α, β) = 0 and p(; α, β) = ( + α), then 0 p(x t ; α, β) for all α [0, ) and β (0, ]. Moveover, the game is not a pure-coordination one (α = 0, β = ), p is always (strictly) concave: p/ x β( + α β) t = [( + α β)x t + β] < 0 x 3 t [0, ], and (α, β) (0, ). Finally, notice that solving for the roots of g(x t ; α, β) = p(x t ; α, β) x t, one gets B = {0, x }, where x (α, β) = β β + α, and 0 x (α, β). We have that x (α, β) = 0 β = and x (α, β) = α = 0. By applying Theorem, points (-3), above one has that only x (α, β) is stable, as p(ε; α, β) > ε for any small ε > 0. Thus Pr{lim t x t = x (α, β)} =. Figure 3 shows some examples of p(x t ; α, β) under a LP rule for dferent values of α and β p(x) α=0.5 β=0.5 α=0. β=0. α=0.0 β=0. α=0.9 β= x Figure 3: An example of p(x t ; α, β) with LP rules for dferent values of α and β. 0

12 Proof of Proposition 4 By replacing expected payoffs () in: one gets p(x t ; α, β) = exp{γ π t (+)} exp{γ π t (+)} + exp{γ π t ( )}, p(x t ; α, β) = and, defining θ = γβ > 0 and σ = α β p(x t ; θ, σ) = α γβ[ + e β x t +x t ], > 0, the more useful two-parameter function: + e θ[σx t +x t ]. Being a logistic function, we know that p C 0 and p/ x t > 0 x t [0, ]. Moreover, p(0; θ, σ) = ( + e θ ) < and p(; θ, σ) = ( + e σθ ) >. Thus, 0 < p(x t; θ, σ) < x t [0, ] and for all (θ, σ) R++. Notice also that p is convex for all x t ( + σ) = β/( α + β) = x and concave for x t x. Hence, at x t = x, p/ x t = 0 for all (θ, σ) R++. Consider first the case α + β = σ =. Here, p(x ; θ, σ) = x =, so that B. However, does not need to be the unique root for g(x t; θ, ) = p(x t ; θ, ) x t = 0. In fact, it is easy to see that is the unique root for g(x t; θ, ) = 0 and only p/ x t xt = θ. In this case, is stable because p is C0 and crosses from above. Conversely, θ >, g(x t ; θ, σ) = 0 will admit two more roots (other than ), say {x(θ), x(θ)}, where it must be that x(θ) < and x(θ) >. Thus, becomes unstable, while x(θ) and x(θ) are both stable for the same argument. Some examples of p(0; θ, ) for dferent values of θ are shown in Figure 4, while Figure 5 plots the roots of g(x t ; θ, ) = 0 against θ: notice how, as θ increases, x(θ) 0 and x(θ). Hence, Theorem applies and the first point of Proposition 4 is proved. Consider now the case α + β σ. Numerical analyses show the existence of two parameter regions, one where the dynamics of the system admits only one stable root, and another one where two stable roots exist, see Figure 6. Notice that when θ, similarly to the σ = case, the system admits only one root x(θ, σ) for all σ > 0. Moreover, as Figure 7 shows, σ < then x(θ, σ) < and decreases as θ increases. Conversely, σ >, then x(θ, σ) > and x(θ, σ) as θ. Finally, x(θ, σ) as σ for any θ, cf. Figure 8. If θ >, Figure 6 shows that the process admits both stable root and stable roots. However, σ > and θ is sufficiently large, then stable roots {x(θ, σ), x(θ, σ)} always emerge. As Figure 9 indicates, in this region x(θ, σ) <, x(θ, σ) >. Finally, as θ, x(θ, σ) 0, x(θ, σ). Once again, by applying Theorem, the second point of Proposition 4 follows.

13 .0 p(x) 0.5 θ = θ > θ < x.0 Figure 4: NP rules. The case σ =. The function p(x t ; θ, ) for dferent values of θ. Figure 5: NP rules. The case σ =. Roots of p(x t ; θ, ) x t = 0 as a function of θ.

14 Figure 6: NP rules. The case σ. Stable roots vs. σ and θ. x* 0.5 σ = 6.00 σ =.50 σ =.50 σ =.0 σ = 0.90 σ = 0.50 σ = θ Figure 7: NP rules. The case σ, θ. The single stable root as a function of θ for dferent values of σ. 3

15 x* 0.5 θ = 0.05 θ = 0.50 θ = 0.0 θ =.00 θ = σ Figure 8: NP rules. The case σ, θ. The single stable root as a function of σ for dferent values of θ. Figure 9: NP rules. The case σ >. The two stable roots vs. θ 5. 4