under Skew-Symmetry Simultaneous Revisions
Tom Norman
Magdalen College, Oxford OX1 4AU, UK
First version: December 4, 2004 This version: September 2, 2008
Abstract Blumes (Games and Economic Behavior 44, 2003, 251271) invariant selection in evolutionary coordination games with skew-symmetric noise processes is extended to the case of simultaneous strategy revisions. JEL Classication: C73. Key Words: evolution; noise; simultaneous revisions; skew-symmetry.
1
Introduction
Stochastic evolutionary models provide a powerful method of equilibrium selection. The concept of stochastic stability (Foster and Young 1990) renes preexisting notions of evolutionary stability by introducing enough noise to overcome path dependence, and then selecting the outcome that is most likely to be observed over the long run when noise is small. And by specifying a xed probability of random mutation or mistake, Kandori, Mailath, and Rob (1993) and Young (1993)henceforth KMRYderive a simple mutation-counting technique for nding a models stochastically stable states. This technique has, however, been criticized by Bergin and Lipman (1996) for its stateindependent mutations, under which agents make mistakes (or experiment) with the same probability irrespective of the current strategy frequencies, and thus of the expected payos at stake. Bergin and Lipman demonstrate that, once mutations are allowed to vary by state, any strict Nash equilibrium of a strategic-form game is selected under some suitably chosen mutation model. Blume (2003) responds to this indeterminacy by delineating the broad class of skew-symmetric noise modelsroughly speaking, those under which only payo dierences, and not the names of strategies, matter to
I am grateful for discussions with David Myatt and Chris Wallace, the helpful comments of the Associate Editor and referee, and the support of All Souls College, Oxford. The usual disclaimer applies. Email [e-mail].
1
choicefor which the usual stochastic-stability results are preserved when there is at most one strategy revision at any point in time.1 Blumes single-revisions assumption (also employed in Binmore and Samuelson, 1997) gives the systems evolution the convenient form of a birthdeath process, sidestepping the sometimes complex tree surgery of the KMRY techniques. Moreover, it can be grounded in an appealing continuous-time setting where the probability of multiple simultaneous revisions is negligible. However, the standard KMR setup allows for simultaneous strategy revisions, and it remains an important case to analyze; aside from its direct interpretation, it is isomorphic to models where other agents actions cannot be observed as frequently as strategy-revision opportunities arise. And the graph-theoretic techniques of Freidlin and Wentzell (1984)upon which the KMRY techniques are basedcan in any case easily cope with Bergin and Lipmans state-dependent mutations. Given KMRs selection of the risk-dominant equilibrium in coordination games irrespective of the speed of the dynamic, one might conjecture that Blumes results too should carry over to the faster simultaneous-revisions setting. This is not obvious, however, given the wide array of transitions available under simultaneous revisions, so that paths between equilibria need not be skew-symmetric even if the noise model is. Nevertheless, the conjecture is conrmed in this note; Blumes (2003) invariant selection resultthat, under random matching and single revisions, the state corresponding to the risk-dominant equilibrium is the unique stochastically stable state for coordination games with skew-symmetric noise processesis extended to the simultaneous-revisions case. This further characterizes the class of models resistant to Bergin and Lipmans (1996) critique of state-independent mutations.
2
Preliminaries
Consider the familiar scenario of a single population of agents repeatedly matched at random in discrete time to play a symmetric 2 2 coordination game with strategy set S = {s1 , s2 } and payo function : S 2 R. The population state z Z records how many of the N agents currently play strategy s1 , and the agents are myopic in the sense that they believe that the current state zt = z will obtain next period (and they are not concerned with subsequent periods). Hence, the expected payo advantage2 (z) :=
1 2
z N z ((s1 , s1 ) (s2 , s1 )) + ((s1 , s2 ) (s2 , s2 )) N N
Blumes analysis is extended by Maruta (2002). Agents can ignore their own presence in the population, given that N is taken to be large.
2
of s1 over s2 is linear and upward sloping, and there is a z (0, N ) such that (z ) = 0, corresponding to the stage games mixed-strategy equilibrium.3 Assume without loss of generality that z < N/2, so that state N corresponds to the stage games risk-dominant equilibrium. The evolution of the system is based on a Markov chain with transition matrix P = (pij ), pij := Prob{zt+1 = j | zt = i}, t N, dened on the nite state space Z and interpreted as the usual behavior of the system. This process is then perturbed by noise to give an irreducible Markov process P = (p )parameterized by such ij that lim P = P which has an ergodic distribution , interpreted as summarizing the long-run behavior of the system. In theory it is possible, given , to compute this invariant distribution simply by solving the stationarity equations P = . However, the practical diculty of this task makes large-deviations techniques useful (see Freidlin and Wentzell 1984, and Young 1998). The following lemma presents the well-known Markov Chain Tree Theorem; let Tz be the family of all trees rooted in a given state z, and p (T ) := (i,j)T p be the likelihood of a particular T Tz . ij Lemma 1 The stationary distribution of the irreducible Markov process P satises (z) = v (z) , iZ v (i) where v (z) =
T Tz
p (T ).
Like Blumes model, the FreidlinWentzell method provides an immediate closed form for the invariant distribution , but it can also cope with simultaneous revisions. Of course, a state z is stochastically stable (Young 1993) if lim (z) > 0.4
3
Skew-Symmetry and Invariant Selection
When only one agent revises his strategy at a time, and his choice obeys some mild payo-sensitivity axioms, Blume (2003) shows that state N is stochastically stable (by virtue of its risk dominance) under any skew-symmetric noise process. Here we provide the formal details for extending this result to the case of simultaneous revisions. Blumes payo-sensitivity axioms require essentially that two strategies with the same payo be chosen with equal probability, and that a strategy be more likely to be chosen the greater its payo advantage over the other. Modifying these axioms for the simultaneous-revisions case, let P satisfy the following assumptions: Axiom 1 piN > 0 and pij = 0, i > z , j < i; pi0 > 0 and pij = 0, i < z , j > i.
Of course, the results apply to any population game with the same (z) function. Blume (2003) also considers relaxations of the assumed linearity of (z) for which his results transfer. 4 It is implicitly assumed that lim (z) exists and is a stationary distribution of P . This requires some (mild) regularity conditions to be placed on the noise modelsee, e.g., Young (1998).
3
3
Axiom 2 As : (a) p = o(p ), i > z , j < j ; ij ij p = o(p ), i < z , j > j ; ij ij (b) p j = O(p ), i > z , i > i > j; ij i p j = O(p ), i < z , i < i < j. ij i Axiom 3 As : p (z +k) = O(p (z +k ) ), |k| > |k |; z z p +k)z = O(p (z +k) ). (z z In general, for two functions f (x) and g(x), if limx |f (x)/g(x)| = 0, then f is of lower order than g, denoted f (x) = o(g(x)); if lim supx |f (x)/g(x)| < , then f is of the order of g, denoted f (x) = O(g(x)); hence, o(g) O(g). In the present context, these measures of order allow us succinctly to capture the relative rates at which transition probabilities vanish as . Axiom 1 says that the transition to the state where everybody plays the higher-payo strategy has positive probability under P (i.e. as noise vanishes), whilst any transition towards the lower-payo strategy does not. In other words, in the absence of noise, there is at least some chance that all agents make the optimal switch each period, but no chance that more agents make the wrong switch than the right switch. A sucient condition for this to hold is that all agents play a best reply each period in the absence of noise.5 Axiom 2(a) says that the order of a transition from a given state is lower the more agents it leaves playing the lower-payo strategy; 2(b) says that the order of a transition to a given state in the direction of the lower-payo strategy is no higher the fewer are the agents originally playing that strategy. The latter part of this is familiar from Blumes axioms, whilst the former part merely extends payo sensitivity to capture the idea that a higher number of mistakes should be less likely than a lower number. Axiom 3 says that, if the two strategies have the same payo, then the order of a transition of k states towards either strategy is nonincreasing in k, and is no lower than the order of the reverse transition. The rst part of this is consistent with both of the main alternative assumptions for transitions away from mixed-strategy equilibria made in the literature: that they have zero probability under P ; or that any such transition has positive probability under P . The second part, meanwhile, is in-keeping with sensitivity payo given the payo disadvantage associated with the switches constituting the reverse transition. Axioms 13 thus constitute fairly mild requirements of payo sensitivity in strategy switches under simultaneous revisions, similar in spirit to those of Blume for the singlerevisions case. Moreover, they are satised by a broad class of models.
Note that, under this axiom, p > 0, i > z (along with p , i < z ), is thus of the same order as i0 iN the (perturbed) probability of any transition with pij > 0. In fact, Axiom 1 can be weakened slightly; all that is required is that any transition to the state where everybody plays the higher-payo strategy is of higher order than any transition towards the lower-payo strategy, and of order no lower than any other transition.
5
4
Example Consider the case of the mistakes model of Kandori, Mailath, and Rob (1993) where agents play a best reply to the current state with high probability (1 ) and the alternative strategy with the complementary low probability > 0. If i > z , then s1 is the best reply so that piN = 1; similarly, pi0 = 1 if i < z . Axiom 1 is thus straightforwardly satised. When the process is perturbed by noise, with parameter = 1/, the transition probabilities become p = ij N N j N j (1 )j
if i > z . Hence, p = O(N j ) as , which is of lower order than p = O(N j ), ij ij i > z , j < j . Also, p j = O(N j ) as , which is of the order of p = O(N j ), ij i i > z , i > i > j. Similar calculations show that the corresponding properties hold when i < z , so that Axiom 2 holds. Finally,
min{z +k,N z }
p (z +k) z
=
m=max{k,0}
z mk
N z m
2mk (1 )N +k2m
= O 2 max{k,0}k = O |k| , as , which is non-increasing in |k|, and is no lower than the order of p +k)z = (z z 0 N z O(1{z ,...,1} (k). +1{0} (k). +1{1,...,N z } (k). ), where 1 is the indicator function; hence, Axiom 3 holds. In addition, the axioms are satised by the stochastic best-reply dynamic employed by Kandori and Rob (1995), and (well-behaved) random-utility models (e.g. Myatt and Wallace 2004).
Blumes skew-symmetry, meanwhile, essentially requires that p i(i+1) = O(pk(k1) ) for all i, k Z such that limN (i) = limN (k)i.e. that the probability of the revising agent switching to s1 for some given expected payo gain g is of the same order as the probability of his switching to s2 for the same gain g. Thus, transition
probabilities are unbiased, in the sense that their order may depend on the expected payo dierence between the two strategies, but not upon the names of the strategies. Under simultaneous revisions, a reformulation of the concept is required in the face of a wider array of possible transitions. Denition 1 P is skew-symmetric if p = O(p j) ) for large N and all i, j, k Z ij k(N such that limN (k) = limN (i). Loosely, the probability of, say, (j i) agents switching to s1 for some given expected payo gain g is of the same order as the probability of (N jk) agents switching to s2 for the same gain g. Or more pertinently, the probability of (N j) agents failing to switch to 5
s1 for some given expected payo gain g is of the same order as the probability of (N j) agents failing to switch to s2 for the same gain g. Intuitively, since it is mistakes that have vanishing probability as , the simultaneous-revisions extension of skew-symmetry should treat transitions with equal numbers of equally large mistakes as equally likely. Note, however, that there will be many other proles of strategy switches eecting the transition i je.g. (j i + 1) agents switching to s1 and one agent switching to s2 (with the remaining agents leaving their strategies unchanged). Thus, there are some aggregation issues to consider before we can claim that this is the appropriate extension of Blumes concept. The transition probabilities under simultaneous revisions have the form p = ij
(i,j)
()Prob ( | (i)) ,
where (i, j) is the set of possible proles of strategy switches and nonswitches realizing the transition i j, and () is the number of possible permutations of (i, j) (essentially relabelling players). Hence, p = O(sup(i,j) Prob( | (i))) ij as . But sup(i,j) Prob( | (i)) will be determined by the number (and probability-order) of mistakes (i.e. actions with vanishing probability) in the transition. If limN (k) = limN (i), then for large N it follows that sup(i,j) Prob( | (i)) = sup(k,N j) Prob( | (k))since the relevant transitions have the same number (and probability-order) of mistakesgiving p = O(p j) ) as in Denition 1. ij k(N Example Consider again our KMR example. If i > z , then p = O(N j ) as . ij If, furthermore, limN (k) = limN (i), then p j) = k(N as required for skew-symmetry. As with the axioms, skew-symmetry will also be satised by many models with state-dependent mutations. With the assumptions in place, we can now put them to use. max {k Z | k z}, and z := min {k Z | k z}. Lemma 2 If P is skew-symmetric, then p = O(p 2z ij (
i)(N j) )
N N j
N j (1 )j = O(N j ),
Let
z
:=
for large N .
Proof. Given linearity of under random matching and the fact that (z ) = 0, it follows that ( 2z i) (i) as N becomes large. The result follows by skewsymmetry. 6
Theorem 1 If P is skew-symmetric, then for all large N the risk-dominant equilibrium state is the unique stochastically stable state. Proof. If z is noninteger, consider an arbitrary T Tz for z { z , . . . , N 1}. Delete the edge (N, j) which begins the path that leads from N to z , and introduce the edge (z , N ). The resulting tree is clearly an N -tree, with p j = o(p N ) by Axiom 1. N z Hence, the new N -tree is of higher order probability than the origi...