Migration and the Evolution of Conventions with Spatially Heterogenous Payoffs (Evolution of Conventions)
Suren Basov Department of Economics University of Melbourne Victoria 3010, Australia
[email protected] February 3, 2006
Abstract
In this paper I consider a model of equilibrium selection in coordination games with mobile players for the case of the spatially heterogenous payoffs. I have shown that for some payoff functions there exists asymptotically stable steady state that does not correspond to a Nash equilibrium for any location, i. e. at any location both strategies are played in the medium-run. I have also shown that both strategies survive in the long-run, i. e. the spatial heterogeneity of the payoffs ensures a richer genetic pool, which makes the population more adaptable to the changing environment. Keywords: Evolution of Conventions, Nonlinear Diffusion Equation, Spatially Heterogeneous Payoffs.
Disequilibrium selection in coordination games and biological advantages of spatial heterogeneity (Disequilibrium selection)
S B
Department of Economics University of Melbourne Victoria 3010, Australia e-mail:
[email protected] tel. (613)-8344-7029 fax. (613)-8344-6899
1
Abstract
In this paper I consider an evolutionary model for 2 × 2 coordination games with mobile players for the case of spatially heterogenous payoffs. I show that for some payoff functions there exists asymptotically stable steady state that does not correspond to a Nash equilibrium for any location, i.e. at any location both strategies are played in the medium-run. I have also shown that both strategies also survive in the long-run, i.e. the spatial heterogeneity of the payoffs results in a richer genetic pool, which makes the population more adaptable to the changing environment. Keywords: Evolution of conventions, nonlinear diffusion equation, spatially heterogeneous payoffs. JEL classification numbers: C73, C65
2
1
Introduction Game theory tries to predict behavior of economic actors in strategic
situations. The main concept is that of a Nash equilibrium, i.e. of a strategy profile such that no player has a profitable deviation. Though the concept of Nash equilibrium embodies the idea of stability of a strategy profile, there are at least two obstacles to using it as a predictive concept. The first, is that many games have multiple Nash equilibria, the second is that it is not clear how exactly do the players arrive at a Nash equilibrium. The evolutionary game theory addresses both of this concerns modelling explicitly the fine-grain adjustments made by the subjects in response to their recent experiences. Usually, these adjustments give rise to a Markov process on the state space, which can be decomposed into a deterministic part (a law governing the change in the expected value of the share of players playing a particular strategy) and a stochastic part. The deterministic dynamics may have multiple asymptotically stable steady states, which are known as the medium-run outcomes, while the Markov process itself is usually ergodic and, therefore has a unique steady state, known as the long-run outcome.1 I will 1
Often the long-run outcome is defined as the limit of this steady state as the stochastic part of the Markov process vanishes.
3
call a model that uses these technical tools a standard evolutionary game model. In this paper I will apply the evolutionary approach to a particular class of games: symmetric 2 × 2 coordination games2 with mobile players. Such games attracted much attention in the literature during the last fifteen years. The pioneering papers in the area, that of Foster and Young (1990), Kandori, Mailath, and Rob (1993), and Young (1993) used evolutionary models with a persistent randomness to study the evolution of conventions. They focused on the models where agents from a homogeneous population interact over time. In these models in the medium-run a single convention prevails (though which one depends on the initial conditions), while in the long-run the riskdominant outcome is selected as the unique stochastically stable solution. This means that, as time goes to infinity, the fraction of time the society will find itself playing the risk-dominant equilibrium goes to one. However, the society will return to the vicinity of the other equilibrium infinitely often. Evolution in these models takes place only in time and the spatial dimension is completely ignored. The main finding of these paper that in the longrun the society finds itself coordinating on the risk-dominant equilibrium for 2
Pure strategy Nash equilibria of such games are known as conventions
4
almost all time. The papers that first introduced some spatial relationship between the players were Anderlini and Ianni (1996), Blume (1993, 1995), Ellison (1993), and Young (1999). In these papers the locations of different players are fixed, the players are connected by some network, and the behavior adopted by an individual depends on its intrinsic payoff and the behavior of her neighbors. This literature confirms the equilibrium selection results described above and shows that the speed of evolution is facilitated by the local nature of the interactions. Seven recent papers by Oechssler (1999), Ely (2002), Mailath, Samuelson and Shaked (2001), Dieckmann (1998), Bhaskar and Vega-Redondo (2004), Anwar (2002), and Blume and Temzelides (2003) allowed individuals to change their location. In all these models the location space is represented by a finite set of “cities.” Over time the players may adjust both their strategy and their location. In Oechssler’s model there are no frictions to the adjustment process. He shows that the efficient convention will eventually prevail provided it was adopted from the start by at least one city. Ely’s model is similar to that of Oechssler. He is able to drop the assumption that the efficient convention
5
is initially present by allowing for mutations. In both models different conventions cannot coexist even in the medium-run, since individuals from the location where the inefficient equilibrium is played will migrate to the location where the efficient one is played at the first opportunity. Dieckmann’s paper shows that the results are robust to the introduction of imperfect information. In all these models an opportunity to revise a strategy arrives at the same moment as an opportunity to migrate. The paper by Mailath, Samuelson and Shaked allows for endogenous interactions, i.e. the players can avoid an undesired match. In addition, in their model time is continuous. They show that the long-run outcome in this case should be efficient. New consensus seemed to arise from this literature. It holds that, in the presence of migration opportunities, Pareto efficient rather than riskdominant outcome is going to be selected in the long-run. Bhaskar and Vega-Redondo challenge the emerging consensus. Contrary to the rest of the recent literature, they assume that opportunities to revise strategy and to migrate never arise simultaneously. They show that the coexistence of the conventions in the medium-run is possible in the pure coordination games. They also show that an inefficient equilibrium can be
6
selected in the long-run. Their equilibrium selection result, however, has nothing to do with the risk-dominance. Anwar looks at the consequences of player movement between locations when there are constraints which limit the number of agents who can reside at each location. He concludes that if the constraints are sufficiently tight then the risk-dominance result continues to hold. However, when sufficient movement is possible, different conventions can coexist in the long-run. Moreover, the location with the tighter constraint coordinates on the payoff-dominant strategy. Blume and Temzelides allow only to some agents to change their location. Both the strategy choice and the location is a myopic best response to the current strategy profile. The strategy choice is subject to noise, while the choice of the location is not. The authors conclude that if mobility of the mobile agents is restricted, the risk-dominant equilibrium is selected at every location. However, if there are enough mobile players to ensure efficient play at some location, then all mobile agents will move to that location. The utility of immobile agents depends on the mobility in a non-monotone way. It increases at the low levels of mobility and decreases otherwise. Note, that migration in this model is strategic, though myopic. The equilibrium
7
selection result crucially depends on the level of mobility within the country. Despite all the differences in modelling assumptions in the papers cited above, the technical tools used by the authors are rather similar. In these sense they are all standard evolutionary game models as described in paragraph two of this Introduction. A paper that stands apart from the most literature is Hobfauer, Hutson, Vickers (henceforth, HHV) (1997). They model the deterministic dynamics by a differential equation for the share of players who play a particular strategy (e. g. the replicator dynamics) and add local non-strategic migration as a diffusion term. Therefore, they obtain a nonlinear partial differential equation (henceforth, PDE) for the share of population playing a particular strategy at a particular location. In this approach the medium-run outcomes correspond to the asymptotically stable steady states of the master PDE, which is similar to the notion of the medium-run outcome in a standard evolutionary game model. The notion of a long-run outcome is, however, somewhat different. In the HHV model the long-run outcome corresponds to the so-called spatial equilibrium. Intuitively, assume that a population is playing a medium-run outcome and a new strategy profile (not necessarily close to the initial one) spontaneously arises in a neighborhood of a particular location. The new strategy profile
8
corresponds to a spatial equilibrium if the bubble of these new strategists will eventually cover the entire population. HHV (1997) proved that thus defined spatial equilibrium coincides with the risk-dominant equilibrium in 2 × 2 symmetric coordination games. Basov (2002) clarified the behavioral foundations of the HHV’s model. It turns out that HHV’s dynamics emerges from a spatial modification of Samuelson’s (1997) aspiration and imitation model, provided that migration occurs much more often than the strategy revision (we travel around the world remaining who we are). Finally, it would be fair to mention the paper of Hansen and Kaarbøe (2002), who looked at the problem of equilibrium selection from a completely different point of view. In a recent paper they argued that if the interaction between players is conducted jointly at a population-wide level and does not lend itself to modelling with pairwise interaction any strict Nash equilibrium may be selected. In this paper I take a closer look at the HHV’s model. First, I prove that within the basic HHV model with spatially homogenous payoffs different conventions cannot coexist in the medium-run and calculate exactly the speed of the switching wave from one convention to another in the case when the deterministic adjustment is the replicator dynamics. I also prove that the
9
speed of the switching wave is proportional to the square root of the openness of economy for any deterministic adjustment dynamics. Though the results stated above are of some interest, they show that apart from a few new insights, the predictions of the basic HHV model do not differ significantly from the other models in the literature. Indeed, in all the papers, including HHV, each location is eventually dominated by one convention. Moreover, with a notable exception of Anwar (2002), one convention eventually dominates globally. The main endeavor of this paper is to challenge these results within a slightly modified HHV model. The main modification is that I allow for the payoffs to differ across different locations. For an example of such situation, assume that you live in a hunter-gathering society and agreed to meet with one of your tribesmen for a deer hunt. You have two choices, to honor the agreement or to break it. If both of you honor the agreement then you will catch the deer and get payoff of one each. If both of you break your word and go fishing (assume it is the second best thing to do) both of you will get payoff zero. If you honor the agreement and your partner breaks his word and goes fishing, he will get zero, while it will be too late for you to go fishing (you are too far from the pond), so you will have to settle for a rabbit. You
10
payoff will in general depend on the density of the rabbit population at the location you leave. I will assume it is −v(x) < 0 (here all the payoffs are net of the energy exerted, so negative payoffs are quite natural). The first important finding of this model is: (R1) For some payoff functions there exists asymptotically stable steady state of the HHV’s dynamics that does not correspond to a Nash equilibrium for any location. To appreciate the importance of this result recall that most of the models in evolutionary game theory are presented as equilibrium selection tools. The deterministic dynamics usually has Nash equilibria as its steady states, while addition of small noise allows to select among different equilibria. This result, on the contrary, states that a non-equilibrium play emerges as a result of the evolution of the players’ behavior. To see that the payoff heterogeneity is crucial for this result recall that for the replicator dynamics without migration any asymptotically stable steady state is a Nash equilibrium. This result remains intact if one introduces migration into a model with spatially homogeneous payoffs. Therefore, simply non-strategic nature of migration is not sufficient to produce a nonequilibrium steady state (as an initial intuition might suggest). It is the
11
interplay between migration and payoff heterogeneity that gives rise to the non-equilibrium steady state. The second important result is: (R2) At each location both strategies are played in the long-run. It is worth noting that the second result is of a particular interest for biology. It states that the genetic variability can be preserved in the population if payoffs differ across locations. This, in turn, can provide a considerable boost to the speed of evolution in a case of a temporal shock to the payoffs. Therefore, species that evolve in a spatially varying environment have a better chance to survive an abrupt environmental change. The paper is organized as follows. In Section 2 I introduce the model of the evolution with migration and study the properties of the basic HHV’s model with the spatially homogeneous payoffs. I show that the medium-run outcome is spatially uniform and is determined only by the spatial mean of the initial distribution, rather than by its finer properties. I also derive the speed of propagation of the switching wave from one convention to another under the replicator dynamics and establish the universal square root law. In Section 3 I introduce spatial dependence of payoff and show that it is possible for an asymptotically stable steady state of the HHV’s dynamics not to correspond to a Nash equilibrium for any location. Moreover, at
12
each location both strategies can be played in the medium-run. Section 4 concludes.
2
The Model with Spatially Homogeneous Payoffs In this Section I am going to describe the model with mobile players
and spatially homogenous payoffs. The model was first introduced by HHV (1997). Its behavioral foundations were established by Basov (2002). It turns out that HHV’s dynamics emerges from a spatial modification of Samuelson’s (1997) aspiration and imitation model, provided that migration occurs much more often than the strategy revision. I will start by describing the HHV’s concept of spatial equilibrium as the long-run outcome of the model, and extend the analysis to study some properties of the medium run outcome. Consider a population of players who dwell in a measurable region Ω of the Euclidean space and are matched to play a symmetric 2 × 2 game. The strategy space of player α is Sα = {X, Y }. In this Section I assume that payoffs are independent of the location and the underlying game is a coordination game. Hence, it has two Pareto ranked pure strategy equilibria
13
(X, X) and (Y, Y ), and a mixed strategy equilibrium. Let u(x, t) be the fraction of the individuals who are located at x at time t and play strategy X. I will assume that x ∈ Rn . In applications n = 1, 2, but I develop the theory for the general case (perhaps, in the anticipation of the space travel). Let φ(u) be the difference in the expected payoffs to strategies X and Y respectively, which depends on the fraction of the individuals playing strategy X and located at x. It is given by:
φ(u) = β(u − u∗ ),
(1)
where β > 0 is some constant that depends on the parameters of the payoff matrix and u∗ ∈ (0, 1) is the fraction of X−strategists that corresponds to the mixed strategy equilibrium of the coordination game. In the absence of migration the evolution of u(x, t) is governed by:
∂u = u(1 − u)φ(u). ∂t
(2)
Equation (2) is known as the replicator dynamics. It states that the rate of change of the fraction of the individuals playing a certain strategy is propor14
tional to the difference between the payoff they earn and the average payoff earned by a representative individual in the population. To introduce migration, consider a compact set V ⊂ Ω with a smooth boundary Σ. The change of the measure of the individuals playing strategy X and located within V occurs due to differential replication and migration. Therefore, assuming that the rate of differential replication is given by the right hand side of (2), one can write
V
∂u dx = ∂t
u(1 − u)φ(u)dx +
V
mdΣ
(3)
Σ
Here vector m is the net migration flow through Σ of the individuals playing strategy X. I assume that the migration decision is not strategic. In that case, the net outflow of the migrants playing strategy X from a particular location will be proportional to the size of the population at that location playing strategy X. Hence, the migration will tend to equalize the fraction of the individuals playing strategy X across the locations, i.e. the migration flow will be proportional to the gradient of u(·):
m = γ∇u.
15
(4)
Coefficient γ ≥ 0 measures the openness of the economy. Small values of γ correspond to a closed economy with strict restrictions on migration. Using the divergence theorem, transform the second term on the right hand side of (3)
mdΣ =
Σ
div(m)dx.
(5)
V
Taking into account that div(∇u) = ∆u
(6)
and that (3) should hold for any compact set with a smooth boundary, one obtains ∂u = f (u) + γ∆u, ∂t
(7)
f(u) = u(1 − u)φ(u)
(8)
where
and ∆u = div(∇u) =
n ∂ 2u i=1
∂x2i
.
(9)
It is easy to introduce strategic considerations into the model. Assume
m = γ[(1 − α)∇u + αβ∇u], 16
where α ∈ [0, 1]. Parameter α measures the strength of the strategic motive. Then u(x, t) will be governed by an equation similar to (7) but with γ replaced with γ ′ = γ(1 − α + αβ). Hence, the model is robust to the introduction of the strategic motive. In particular, the equilibrium selection result survives it. To proceed with a formal analysis of equation (7) let us define a function
V (u) =
u
f (z)dz,
0
and a functional F (u) =
γ [ ∇u 2 − V (u)]dx, 2
(10)
Ω
where · denotes the Euclidean norm of a vector. I will begin the analysis of dynamics (7) proving the following theorem: Theorem 1 Let equation (7) hold. Then
dF (u(t)) ≤ 0. dt
17
(11)
Proof. Note that equation (7) can be rewritten in the form
δF ∂u =− , ∂t δu
(12)
where δF/δu is the variational derivative of F (for a definition of the variational derivative see, for example, Gelfand and Fomin, 2000, p. 28). Then
dF (u(t)) δF ∂u ∂u = = −( )2 ≤ 0. dt δu ∂t ∂t
(13)
Theorem 1 by itself is a purely technical result. It can be easily extended to the case of spatially heterogenous payoffs. The main role of Theorem 1 is that it enables us to analyze the stability of certain solutions. In particular, a following corollary holds. Corollary 2 Let Ω have finite positive Lebesgue measure and let u∗ be a strict local maximum of V (·) Then a stationary uniform solution of (7)
u(x, t) = u∗
is locally asymptotically stable. 18
(14)
Proof. Define function L(·) by
L(t) = F (u(t)) +
V (u∗ ) , µ(Ω)
(15)
where µ(Ω) is the Lebesgue measure of the region populated by the players. Then L(t) =
γ [ ∇u 2 + (V (u∗ ) − V (u))]dx ≥ 0, 2
(16)
Ω
with strict inequality if u differs from u∗ on a set of a positive Lebesgue measure. Moreover, Theorem 1 implies that
dL dF = ≤ 0, dt dt
(17)
with strict inequality for any u such that f (u) (and therefore, ∂u/∂t) is not zero. Therefore, L(·) is a Lyapounov function for (7) and u∗ is asymptotically stable. Since the Nash equilibria can be found from the condition
V ′ (u) = 0,
(18)
function V (·) can be interpreted as a potential of the game. Below, I will 19
argue that it can also be interpreted as the stochastic potential, in the sense of Foster and Young (1990). I will call the local maxima of V (u) the medium-run outcomes. Note that though all medium-run outcomes are Nash equilibria of the underlying coordination game the reverse is not true. Apparently, a medium-run outcome that will eventually be reached by the population is history dependent.
2.1
The Long-Run Outcome
In this subsection I study the spatially stable solution of equation (7), which I refer to as the long-run outcome and the dynamics of convergence to it. The spacial stability of the risk dominant equilibrium, was first proved by HHV (1997). Their proof works for any learning process, however it does not allow to calculate explicitly the speed of propagation of the switching wave. The contribution of this subsection is twofold. First, I assume that the learning process can be captured by the replicator dynamics and provide a different proof of the stability of the risk dominant equilibrium, which as a by-product allows us to arrive at the speed of propagation of new conventions. I also establish that the speed of propagation of new conventions is proportional to the square root of openness of the economy for any deterministic 20
dynamics and estimate the life tine of the risk-dominated convention. Second, I demonstrate that while in the one-dimensional world the risk-dominant convention will spread as soon as sufficient fraction of population at a given location switches to it, in a two-dimensional world it requires some area of a minimal size to switch to a new convention before it can spread. To begin the formal analysis as a first observation note that function V (·) achieves its local maxima for the monomorphic populations (u = 0 or u = 1). Indeed, from the definition of V (·):
V ′ (u) = βu(1 − u)(u − u∗ )
(19)
V ′′ (u) = β(2(1 + u∗ )u − 3u2 − u∗ ).
(20)
V ′ (0) = V ′ (1) = 0,
(21)
V ′′ (0) = −u∗ < 0
(22)
V ′′ (1) = u∗ − 1 < 0.
(23)
Therefore,
According to Corollary 1 both monomorphic states are locally stable. As-
21
sume, however, that in an initially monomorphic population once in awhile some non-trivial fraction of the population mutates, i.e. switches to an alternative strategy. A sufficiently big exogenous disturbance of this kind can take the population away from a local maximum to the global one. A steady state, which delivers the global maximum to V is called stable, while the other local maximum is called metastable. Let us prove the claim made in the previous paragraph, i.e. establish that in the long-run the population will switch from the metastable steady state to the stable one. In the process of demonstration we will see that the switching does not occur simultaneously at all locations, but rather travels along the population in a form of a switching wave. Formally, once has to establish that equation (7) has a particular kind of solutions, known as wave solutions. Let us first consider the case n = 1 and look for a solution of equation (7) of the form:
u = u(x − ct),
22
(24)
satisfying the boundary conditions:
lim u(y) = 0, lim u(y) = 1
y→∞
(25)
y→−∞
where y = x − ct. Substituting (24) into (7) one obtains
γu′′ = −f (u) − cu′ .
(26)
Now multiply both sides of this equation by u′ and integrate from −∞ to ∞ to arrive at:
γ
+∞ −∞
u′ (y)u′′ (y)dy = −
+∞ −∞
f(u(y))u′ (y)dy − c
∞
[u′ (y)]2 dy.
(27)
−∞
Let us try to find the solution that additionally satisfies:
lim u′ (y) = 0.
y→±∞
(28)
These condition will be automatically satisfied if u(·) approaches its boundary values monotonically. Taking into account the boundary conditions (25), (28)
23
and the explicit expression (8) for f (u): +∞
u′ (y)u′′ (y)dy =
−∞ +∞
′
f (u(y))u (y)dy =
−∞
1 ′ [u (y)]2 |+∞ −∞ = 0 2
(29)
1
(30)
f (u)du = −
β (1 − 2u∗ ) 12
0
and therefore: c
∞
[u′ (y)]2 dy =
−∞
β (1 − 2u∗ ). 12
(31)
Equation (31) implies that if c > 0 then there exists a switching wave that starts at equilibrium (Y, Y ) and moves the population to equilibrium (X, X). In that case (X, X) is called a spatial equilibrium. Otherwise, (Y, Y ) is the spatial equilibrium. Intuitively, the spatial equilibrium is an asymptotically stable steady state that is also stable against finite perturbations, i.e. the mutation that makes a finite set of players at a neighborhood of particular location to change their strategies. Note that c > 0 if and only if u∗ < 1/2. Therefore, the spatial equilibrium coincides with the risk-dominant equilibrium in the sense of Harsanyi and Selten (1988). For a special case when c = 0 (u∗ = 1/2) the spatial
24
equilibrium is given by a standing wave
u(x) = 1 + exp(
1
β (x 2γ
,
(32)
− x0 ))
where x0 is an arbitrary constant that reflects that with spatially homogeneous payoffs any point can serve as the origin. This case is, however, not generic. Moreover, as I will show in the next subsection, this solution is not asymptotically stable. To find the value of c let us introduce a notation
p(u) = u′ (y).
(33)
Function p(·) gives the derivative of u with respect to y as a function of u at point y. Differentiating the above formula with respect to y one obtains:
u′′ = p′ (u)u′ = p′ (u)p(u).
25
(34)
Therefore, population (25)-(26) can be written as:
γpp′ = −βu(1 − u)(u − u∗ ) − cp
(35)
p(0) = p(1) = 0.
(36)
Let us look for its solution in the form
p = Au(u − 1).
(37)
γA2 u(u − 1)(2u − 1) = βu(u − 1)(u − u∗ ) − cAu(u − 1),
(38)
γA2 (2u − 1) = β(u − u∗ ) − cA.
(39)
Then
or
Since (39) should hold for any u, the coefficients before u and the free term on the right hand side and the left hand side should be equal. Therefore,
2γA2 = β
γA2 = 2γA2 u∗ + cA
and solving for c one obtains
26
(40)
c=
βγ (1 − 2u∗ ). 2
(41)
Finally, the explicit solution of (24)-(25) is given by:
u(x, t) = 1 + exp(
1 β (x 2γ
,
(42)
− x0 − ct))
where x0 is an arbitrary constant. Note that this solution satisfies conditions (28). One can verify by a direct calculation that c > 0 if and only if V (0) > V (1), hence in the long run the population will adopt a convention with a higher value of V . The above analysis shows that generically the long-run outcome is unique. However, if V (1) = V (0) then c = 0 and both steady states can coexist in the long-run. At the point of switching u will change with a jump. One can also see that c > 0 if and only if u∗ < 1/2. This implies that if the society was in the risk-dominant equilibrium any mutation that affects a subset of the population with the measure less then one will eventually die out. Hence, assuming the probability of a mutation affecting almost all the population is zero, the society will remain at the vicinity of the risk-dominant 27
equilibrium forever with probability one, once this equilibrium is reached. On the other hand, if the society started from the risk-dominated equilibrium a mutation has a chance of spreading and driving it to the risk-dominant one, provided it is big enough. It is worth noting that function V (·) coincides with the stochastic potential of Foster and Young (1990). Formula (41) implies that the speed of the switching wave is increasing in the degree of openness of the economy and is proportional to the square root of the openness. This conclusion is universal. Universality here has a double meaning. First, it does not depend on the functional form of φ(·), i.e. it will still hold if replicator dynamics in (7) is replaced by some other payoff monotone dynamics. Second, it does not depend on the dimensionality of space. To prove these assertions, let us look for a solution of (7) in a form
u(x, t) = u(ξ) ξ =
x − ct √ . γ
(43) (44)
Then (7) implies c √ u′ = −f (u) − ∆u. γ
(45)
The speed c can be determined from (45) subject to (25). Since neither right 28
hand side of (45) nor (25) contain γ, solution to (45)-(25) will have a form
c=
√ γh(α, [f ], [φ]).
(46)
If one hypothesizes that the globalization increases the degree of openness, it would imply that the globalization will increase the speed of the universalization of the behavior across the different locations. So far, I assumed that the population moves from one steady state to another and computed the speed of the switching wave. Next, I will ask what is the minimal disturbance that will bring a population from one steady state to another. More specifically, assume that the globally stable state is u = 1, that is (X, X) is the risk dominant equilibrium and u∗ < 1/2. Then 1
f (u)du > 0.
(47)
0
Assume also that at t = −∞ the population is at the state u = 0. Then at
29
time t = 0 a disturbance centered near location x = 0 is created, that is u(x, 0) = u0 , 0 < u0 ≤ 1 ∂u . (x, 0) = 0 ∂x lim|x|→∞ u(x) = 0
(48)
One can interpret this population as describing an invasion of mutants that settle near x = 0. Note that I allow for u to decrease arbitrary fast, that is an invasion can be local. The question is, what is the minimal value of u0 that will cause switching to the state u = 1. The minimal disturbance u corresponds to the stationary unstable solution of (7). Using (43)-(44) one can reduce (7) to: γu′′ + f(u) = 0
(49)
subject to (48). Integrating (49) with respect to x and using (48) one obtains: u0
f (u)du = 0.
(50)
0
The explicit solution of (50) satisfying u0 ≤ 1 is given by: 1 u0 = (2 + 2u∗ − 3
30
1−
9u∗ ). 2(1 + u∗ )2
(51)
As u∗ changes from zero to 1/2, u0 increases from 2/3 to 1. That is, the minimal fraction of mutants at a particular location needed to initiate the transition is 2/3. However, for n = 1 the mutation can be totally localized. Since 2/3 > 1/2 ≥ u∗ , the transition to the risk-dominant equilibrium at a fixed location takes longer then in a model without migration opportunities, but once sufficiently big local disturbance arises at spreads relatively fast all over the world. More precisely, if the integral in (50) is large enough, that is the value of V (·) at the global maximum is much bigger than at the local one, u0 becomes smaller and c becomes very big. This implies that in this case a smaller mutation is sufficient to bring the population from a metastable to the stable set and it travels fast. Vice versa, if V (·) achieves approximately the same value at both local maxima, then a large mutation is needed to take the population from a local to the global maximum and it travels slow. To estimate the life-time of the risk-dominated convention, assume that the probability of that a disturbance of size z covering length (area) s will occur during a time interval τ is
q = (1 − exp(−
31
zτ ))(1 − s) ν
for some ν > 0. Note that we assume that the probability of a global mutation is zero. This assumption assures that once the risk-dominant equilibrium is reached the society will remain in its vicinity forever with probability one. Then the time it takes for the whole population to converge to the long-run outcome is of the order of T ∼
u0 1 + . ν c
The first term here is the time necessary for a sufficiently big disturbance to arise, while the second is the time it takes for such a disturbance to spread all over the world. Note that T does not depend on the population size. It means that new conventions can take hold rather quickly even in a big population. This result is, of course, not surprising in the light of the previous literature on the local interactions. Now let us briefly consider the case n = 2. Introduce the polar coordinates (r, θ) on the plane by:
x1 = r cos θ x2 = r sin θ,
where r is the distance from point x to the center of the switching wave and 32
φ is the polar angle. Symmetry suggests that u = u(r). Let c(R) be the speed of propagation of the circular switching wave with radius R. Then, repeating the derivation that lead to equation ((26)), one obtains:
γ −(c(R) + )u′ = f (u) + γu′′ . r
If the boundary that separates areas with different conventions is thin, then c(R) is approximately given by
c(R) = c −
γ , R
(52)
where c is the speed of propagation of the switching wave in the one-dimensional population with the same payoff function. For the details, see Markstein (1951). Note, that new conventions spread slower in the two-dimensional world than in the one-dimensional one. Moreover, for
R
1 is straightforward. If n = 1, without loss of generality, assume Ω = [0, 1]. Let us expand u(x, t) in the Fourier series
u(x, t) =
∞
un (t) exp(2πinx)
(54)
k=−∞
where i is the imaginary unit (i2 = −1), and un (t) is defined by
uk (t) =
1
u(x, t) exp(−2πinx)dx.
(55)
0
Substituting (54) into (7) and collecting the terms before the same exponents results in the following infinite population of the ordinary differential
35
equations:
du0 = f(u0 ) dt
(56)
du3k±1 = −4γπ 2 u3k±1 − u∗ u3k±1 dt du3k±2 = −4γπ 2 u3k±1 − u∗ u3k±2 + (1 + u∗ )u23k±1 dt du3k±3 = −4γπ 2 u3k±3 − u∗ u3k±3 + u33k±1 dt
(57) (58) (59)
for any k ∈ Z. Equation (56) implies that u0 (t) will converge generically to one of the local maxima of V (u0 ). Moreover, the local maximum to which u0 (t) will eventually converge depends only on u0 (0). Equation (57) implies that u3k±1 converges to zero, but then equations (58) and (59) imply that u3k±2 and u3k±3 converge to zero as well. Hence, any initial distribution of the strategies converges to a spatially uniform stationary outcome. Since, according to (55), u0 (t) is the spacial average of u(x, t) at time t, the mediumrun outcome is determined by the spacial average of the initial distribution only. Note that the medium-run outcome is spatially uniform, provided payoffs do not depend on x. This implies that for customs and conventions to differ across the locations (for example, for the national cultures to exist) in the
36
medium-run one has to postulate that the payoffs for some strategy profile differ across the locations.
3
The Model with Spatially Heterogenous Payoffs In this Section I modify the basic model to include the spatial dependence
of payoffs. Normalize payoffs in the following way:
π(X, X) = 1
(60)
π(Y, X) = π(Y, Y ) = 0
(61)
π(X, Y ) = −v(x) < 0.
(62)
Let u∗ (x) be the probability with which strategy X is played in the mixed strategy equilibrium. Then u(x, t) is governed by the following nonlinear PDE: ut = u(1 − u)(u − u∗ (x)) + γ∆u(x).
(63)
In the next two subsections I am going to provide an example of an asymptotically stable nonhomogeneous steady state that does not correspond to a 37
Nash equilibrium at any location and discuss its the typicality.
3.1
An example of a disequilibrium asymptotically stable steady state
In this subsection I am going to provide an example of an asymptotically stable nonhomogeneous steady state that does not correspond to a Nash equilibrium at any location. .Let n = 1, γ = 1/4 and
u∗ (x) =
1 − tanh x , 2
(64)
where the hyperbolic tangent is defined by
tanh x =
exp(x) − exp(−x) . exp(x) + exp(−x)
(65)
This corresponds to v(x) = exp(−2x).
38
(66)
Note that equilibrium (X, X) is risk-dominant for x > 0, while (Y, Y ) is risk-dominant for x < 0. It is straightforward to check that
u0 (x) =
1 + tanh x 2
(67)
satisfies (63). The graph of this function is depicted on the Figure 1. Insert Figure 1 here. Note that the solution converges to the risk-dominant equilibrium as the absolute value of x goes to infinity. However, for any finite x both X and Y strategists are present at the location. Moreover, there share does not correspond to any equilibrium share apart from point x = 0. Let us prove that this solution is asymptotically stable. For this purpose, for any u ∈ C 1 (R × R++ ) define a function
V (u, x) =
u
f (z, x)dz,
0
and a functional F (u) =
1 [ ∇u 2 − V (u, x)]dx. V 8
39
(68)
For any h ∈ H 1 (R++ ) define
G(τ ) = F (u0 + τ h).
(69)
Using the same logic as in the proof of Theorem 1 one can show that G(·) decreases along the solutions of (63). Therefore, to prove that u0 is asymptotically stable it is sufficient to prove that G(·) achieves a local minimum for τ = 0. By construction, G′ (0) = 0 and ∞
1 G (0) = 2 ′′
(h2x −
1 − 5 tanh2 x 2 h )dx. 4
(70)
−∞
To prove that G′′ (0) > 0 it is sufficient to prove that
min h
∞
(h2x
−∞
1 − 5 tanh2 x 2 − h )dx > 0, 4 s.t.
∞
h2 (x)dx = 1.
(71)
(72)
−∞
The last requirement is due to the fact that G′′ (0) is homogenous in h. The minimum is equal to the smallest eigenvalue of the linear differential operator
L=
d2 1 − 5 tanh2 x + , dx2 4 40
(73)
i.e. we have to find the minimal λ such that the following boundary problem has a solution
hxx +
1 − 5 tanh2 x h = λh 4 lim hx (x) = 0.
|x|→∞
(74) (75)
Following Landau and Lifshitz (1958), introduce √ 21 − 1 λ = E + 4, s = 2
(76)
and let us make the following change of variables
w coshs x
(77)
y = sinh2 x,
(78)
h =
where the hyperbolic sine and cosine are defined by
exp(x) − exp(−x) 2 exp(x) + exp(−x) cosh(x) = . 2 sinh(x) =
41
(79) (80)
Then equation (74) is reduced to
1 1 y(1 + y)wyy + ((1 − s)y + )wy + (s2 + E)w = 0, 2 2
(81)
This is a hypergeometric equation, which has a finite solution only if E < 0 and s−
√ E = n,
(82)
where n is a natural number. Therefore, the smallest eigenvalue of operator L is
√ 21 − 1 2 ) ≈ 0.79 > 0. λ0 = 4 − ( 2
(83)
Therefore, u0 (x) is asymptotically stable. Moreover, the relaxation time towards it provided the initial state is in its basin of attraction is of the order 1/λ0 . Moreover, evaluating functional F (·) one can show that
F (u0 ) = −0.07129 < F (0) = F (1) = 0,
(84)
which implies that the long-run outcome is not spatially uniform. (Note that the integral that defines F (1) can be calculated only in the terms of principal value). However, we cannot claim that u0 is the long-run outcome, since the 42
model can have other spatially non-uniform solutions with lower values of F (·).
3.2
Are disequilibrium asymptotically stable steady states typical?
In the previous subsection I provided an example of a disequilibrium asymptotically stable steady state reached by a population that played a coordination game with spatially heterogenous payoffs. In this subsection I am going to address the questions: Are such solutions a result of a fine-tuning of the parameters of the model or are they quite typical? The first step in answering this question is to establish the existence of stationary spatially non-uniform solutions of equation (7). It turns out that existence can be established under rather general condition on both finite and infinite domains (see, for example, Bernfeld and Lakshmikantham, 1974). Let u0 (x) be such a solution. To analyze its stability one has to study the eigenvalues of the linear differential operator
H =−
1 d2 + V (x), 2 dx2
43
(85)
where V (x) =
2 fu (u0 (x), x). γ
(86)
The solution is stable if and only if all eigenvalues of operator H are negative. Operators of form (85) are known as Schrödinger operators. They are well studied and operators with negative eigenvalues are by no means exceptional. (see, for example, Landau and Lifshitz, 1958). Therefore, the situation described in the example is rather typical.
4
Conclusions In this paper I extended the HHV’s (1997) model for the case of the spa-
tially nonhomogeneous payoffs. I have shown that for some payoff functions there exists asymptotically stable steady state of the HHV’s dynamics that does not correspond to a Nash equilibrium for any location and argued that it is the interplay between migration and payoff heterogeneity that gives rise to the non-equilibrium steady state. I also showed that for this steady state at each location both strategies are played in the medium-run. Moreover, this result remains valid even in the long-run. Though a possibility of coexistence of conventions in the medium-run is known in the literature, it is always a
44
coexistence at different locations. In this model we obtain a possibility of coexistence at the same location. From a biological perspective, the second result states that the genetic variability can be preserved in the population if payoffs differ across locations. This, in turn, can provide a considerable boost to the speed of evolution in a case of a temporal shock to the payoffs. Therefore, species that evolve in a spatially varying environment have a better chance to survive an abrupt temporal change.
45
REFERENCES
L. Anderlini and A. Ianni, 1996, Path dependence and learning from neighbors, Games and Economic Behavior, 13, 141—177. A. W. Anwar, 2002, On the coexistence of conventions. Journal of Economic Theory, 107, 145-155. S. Basov, 2002, Evolution of Social Behavior in the Global Economy: The Replicator Dynamics with Migration, The University of Melbourne, Department of Economics, Research Paper #847, http://www.economics.unimelb.edu.au/research/workingpapers/wp02/847.pdf S. R. Bernfeld and V. Lakshmikantham, 1974, An introduction to nonlinear boundary value problems (Academic Press, New York, NY). V. Bhaskar and F. Vega-Redondo, 2004, Migration and the evolution of conventions. Journal of Economic Behavior and Organization, 55(3), 397-418 L. Blume, 1993, The statistical mechanics of strategic interaction. Games and Economic Behavior 5: 387-423. L. Blume, 1995, The statistical mechanics of best response strategy revision. Games and Economic Behavior 11: 111-145. A. Blume and T. Temzelides, 2003, On the geography of conventions. Economic Theory 22, 863-873. 46
T. Dieckmann, 1998, The evolution of conventions with mobile players. Journal of Economic Behavior and Organization, 38, 93-111. G. Ellison, 1993 Learning, local interaction, and coordination. Econometrica 61, 1047-1071. J. Ely, 2002, Local conventions. Advances in Theoretical Economics, 2: 1-30. D. Foster and P. Young, 1990, Stochastic evolutionary game dynamics. Theoretical Population Biology, 38: 219-232. I. M. Gelfand and S. V. Fomin, 2000, Calculus of Variations, (Courier Dover Publications, Mineola, NY). P. S. Hansen and O. M. Kaarbøe, 2002, Equilibrium selection in coordination games with simultaneous play. Economic Theory, 20: 793-807. J. Harsanyi, and R. Selten, 1988, A general theory of equilibrium selection in games, (MIT Press, Cambridge, MA). J. Hofbauer, V. Hutson, and G. T. Vickers, 1997, Travelling waves for games in economics and biology, Nonlinear Analysis, Theory, Methods, and Applications, 30, 1235-1244. M. Kandori, G. Mailath, and R. Rob, 1993, “Learning, Mutation and Long Run Equilibria in Games,” Econometrica, 61, 29-56.
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L.D. Landau and E.M. Lifshitz, 1958, Quantum mechanics, non-relativistic theory. ( Pergamon Press: London). G. Mailath, L. Samuelson, and A. Shaked, 2001. Endogenous interactions. In: Nicita A., Pagano. U (eds.) The evolution of economic diversity. (Routledge: New York). G. H. Markstein, 1951, Experimental and Theoretical Studies of Flame Front Stability, Journal of Aeronautic Science, 18, 199-201. J. Oechssler, 1999, Competition among conventions. Mathematical and Computational Organization Theory, 5: 31-44. L. Samuelson,1997, Evolutionary games and equilibrium selection. (MIT Press: Cambridge, MA). P. Young, 1993, The evolution of conventions, Econometrica, 61, 57-84. P. Young, 1999, Diffusion in social networks, Center on Social and Economic Dynamics, Working Paper #2.
48
1
y
0.75
0.5
0.25
0 -2.5
-1.25
0
1.25
2.5 x
Figure 1
49
