Conventions

Aconvention, see Young [1993], is an expected, possibly non-symmetric equilibrium. Young [1993] looks at an evolutionary framework to examine how these arise and how stable they are. The basic model consists of an n person game, with players drawn from large but finite population, players basing actions on sample of events from recent past and occasionally mistakes/experiments.

This is similar to fictitious play (see section 2.6.2) but with limited memory. For the basic form of adaptive play there need not be convergence to a Nash equi- librium (pure or mixed); but when we have stochasticity and finite memory we have a stationary distribution in the long run; if all of the weight of this distribution is on one equilibrium we call itstochastically stable.

Formally let Γ be n person game in strategic form, with Si the finite set of

strategies available to player i. N is population of individuals partitioned into n

non-empty Cj. Each member of Cj could play role i in game. All individuals in a

class have same utility functionui(s) for s∈S, set of strategy tuples. t = 1,2, . . .

are successive time periods. Player i chooses pure strategy si(t). Let combined

tuple be s(t) and history up to t, h(t). Each player takes k samples from m most recent periods to determine strategy. So now think of general historyhas a state in Markov chain, with successor obtained by deleting leftmost element and adjoining new rightmost element. We have transition rule which is a best reply distribution, that is give positive probability to strategy if and only if ∃ sample to which it is best reply (and also is independent oft). Define:

P_hh0 0 = Y

i

pi(si|h).

whereP_hh0 0 = 0 if h0 not a successor state of h. Clearly a strict, pure Nash equilib-

rium. m times in succession is an absorbing state, call this a convention. Conver- gence to absorbing state implies a strict, pure Nash equilibrium.

We can define a best reply graph of Γ such that each vertex s is n-tuple of strategies and there is directed edges→s0 if and only ifs6=s0 and exists an agent

graph has no directed cycles. Weakly acyclic if for all initial vertices s, there is a directed path tos∗ from which there is no exiting edge. L(s) is length of shortest path froms to a strict Nash equilibrium LetLΓ= maxsL(s).

Young obtains the result that if Γ a weakly acyclic n-person game. If k ≤

m/(LΓ+ 2) then adaptive play converges almost surely to a convention25.

Now we introduce noise term - probability a player in roleiexperimentsλi.

Now letqi(s|h) be conditional probability thatichoosessgiven that he experiments

and history h. It turns out that subject to suitable conditions selected equilibria are independent of precise q, λ. The conditional transition probability for a subset of playersJ is QJ_hh0 = Y j∈J qj(sj|h) Y j / ∈J pj(sj|h) or QJ_hh0 = 0−ifh0 not successor

and new perturbed Markov process has transition function:

P_hh 0 = ( n Y i=1 (1−λi))Phh0 0+ X J⊆N,J6=∅ |J|(Y j∈J λj)(( Y j /∈J (1−λj))QJhh0.

We call this process adaptive play with memory m, sample size k, experimentation probabilitiesλi and experimentation distributionsqi. P0is the unperturbed process.

Now consider P _{where is small. This is irreducible and aperiodic, so has}

unique stationary distributionµ (that is µP =µ). Letµ_h be probability thath

is observed at any sufficiently larget. A state isstochastically stable relative to P

if lim→0µh >0. Define a mistake forh0 successor ofh as component of si of snot

an optimal response byi to any sample of sizek from h. The resistance r(h, h0) is total number of mistakes in transitionh→h0 (if h0 successor) or∞ (otherwise).

Now view H as vertices of directed graph. For every h, h0 insert directed edge if r(h, h0) is finite and let this be it’s weight. Let H1, H2. . . , HJ be recurrent

communication classes ofP. Now look at graph G with vertices Hk and directed

edges with weightri,j, the path of least total resistance between classes.

Finally for every vertex letT be set off alli-trees26onG. The resistance ofi- tree is sum of resistances of edges. Thestochastic potential of Hi is least resistance

among all i-trees. Can obtain the result that the stochastically stable states of adaptive playP are states in recurrent communication classes ofP0with minimum stochastic potential.

25_{This bound might not be tight.} 26

This kind of model can be developed and applied in many directions. In Young [1998] the above ideas are developed further via the introduction of a spatial model (in Chapter 6), looking at more generaln-person games (in Chapter 7) and bargaining (in Chapter 8). which includes the introduction of a limited sort of heterogeneity. In Axtell et al. [2001] Axtel, Epstein and Young explore a model of the evolution of conventions where certain members of the population have “tags” (e.g. black, white) but are otherwise identical. While an equitable norm is the only stochastically stable norm, a discriminatory norm may persist for many periods. The concept they use to describe this is that of broken ergodicity: the process is technically ergodic but may spend a lot of time in a far from asymptotic state. In Young [2009] a simple framework for diffusion (where agents have a propensity to adopt) is explored via three mechanisms: contagion, social influence and social learning. This method will be used in chapter 6 in order to analyse the long run behaviour of a set of economic models.