Chapter 53 in Handbook, Plott & Smith(2008)

EQUILIBRIUM CONVERGENCE IN NORMAL FORM GAMES

NICOLE BOUCHEZ and DANIEL FRIEDMAN

Economics Department University of California, Santa Cruz

Figure 3. Single population sessions: Exp3 runs 1每4. These graphs chart the time path of pt in the first four runs of the first usable session. The type 1 payoff matrix here has unique mixed NE p∗ = 2/3 = 8/12. That is, in NE 8 of 12 players choose the first strategy (or 4 of 12 when the matrix rows and columns are interchanged as in runs 2 and 3). The graphs show a tolerance of 1 player in the band around NE. The time paths in the first four runs suggest that the NE attracts states pt outside the tolerance band p∗ ㊣ 1/12, but there is considerable behavioral noise so hits occur in only about 50% of the periods.

Figure 4. Two population sessions: Exp5 runs 1每4. Graphs of the behavior in the
first four runs of exp5, the first 2-population session. All periods use the
buyer每seller matrix or its interchange, so the unique NE (denoted by Q) is at
(p, q) = (.25, .50) or, for the interchanged version, at (.75, .50). The graphs
show a 2-period moving average of the time path in the unit square. The graph
for the first run looks like an unstable counterclockwise spiral diverging from
the NE. The second run looks like a tidy counterclockwise double loop around the
NE, neither converging nor diverging. The third run uses the RP matching
protocol; at best there is a weak tendency to drift towards the NE. The fourth
run reverts to MM and looks like a counterclockwise spiral possibly converging
to the NE.

A Comparison of Learning and Replicator Dynamics Using Laboratory Data //
1998-11-22
Dan Friedman , Yin-Wong Cheung

Abstract We compare the explanatory power of replicator dynamics with belief learning dynamics using laboratory data in a variety of games. Overall, and especially in two-population games, the belief learning model fits the data better.

A Comparison of Learning and Replicator Dynamics Using Laboratory Data

Abstract We compare the explanatory power of replicator dynamics with belief learning dynamics using laboratory data in a variety of games. Overall, and especially in two-population games, the belief learning model fits the data better.

A Comparison of Learning and Replicator Dynamics Using Laboratory Data

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Figure 5. Single population three choice sessions: Exp57 runs 1, 4, 6, 7. Graphs of the behavior in the first four HDB runs of exp57. The two NE are at (p, q) = (0, 0) and (2/3, 1/3) (for the interchanged version, at (1, 0) and (0, 1/3)) and are represented by 2 and Q, respectively. The graphs show a 2-period moving average. The run 1 has loose convergence in the second half of the run as does run 7. Run 4 shows no convergence in either half. Run 6 has loose convergence in both the first and second halves of the experiment. (The data are sparse but consistent with evolutionary game theory.)

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2. Results

We have collected more than 300 such runs and used various statistical tests as
well as

summary graphs to study convergence properties. Figures 3每5 show some of the
summary

graphs for both one, two population, and three choice games. The main findings,

presented more fully in Friedman (1996), can be summarized as follows.

(1) Some behavioral equilibrium (BE) is typically achieved by the second half of

a 10 to 16 period run. The operational definition of BE is that strategy
selection

is almost constant in each population in a given run (or half-run). ※Almost

constant§ means that the mean absolute deviation from the median number of

players choosing a given strategy is less than 1 player (※tight§) or less than 2

players (※loose§). Overall we observe tight BE in over 70% of second half-runs

and loose BE in over 98% of second half-runs. Tight BE was achieved most
reliably

in type 3 games (over 95% of all half-runs). By contrast, type 1 games

achieved tight BE in only 55% of half runs, but achieved loose BE in 96%.

(2) BE typically coincides with a Nash equilibrium (NE), especially with those

(called evolutionary equilibria (EE)) that evolutionary game theory identifies
as

stable. We operationalize NE by replacing the median number of players by the

NE number. In second half-runs, for example, about 79% of the loose BE are

loose NE, and 84% of those are loose EE. The only notable exception to this

conclusion is that in type 2 runs the BE sometimes coincided with the non-EE

mixed NE. A closer look at the graphs suggests that many of these cases actually

represent slow divergence from the mixed NE, and many of the half-runs

deemed BE but not NE seem to represent slow or incomplete convergence to an

EE (a pure NE).

(3) Convergence to BE is faster in the mean-matching (MM) than in the
randompairwise

(RP) treatment, and faster in the Hist treatment than in the No Hist

treatment. In particular, the slow and incomplete convergence observed in type-1

games arises mainly in RP matching protocol and No Hist runs. The results from

type-2, single population games and all two-population games support the same

conclusion. There is, however, an interesting exception. The few instances of

non-convergence in type-3 games arise more often under MM than under RP.

(4) Individual behavior at a mixed strategy BE is better explained by
idiosyncratic

※purification§ strategies than by identical individual mixed strategies. In
particular,

in the simplest type 1 game, Hawk每Dove, we see persistent heterogeneity

in which some players consistently pick the first (※Hawk§) strategy and others

consistently pick the other (※Dove§) strategy.

(5) ※Hawk每Dove每Bourgeois§ is a 1-population 3-action game with a triangular
state

space and with one corner NE (an EE) with target area b2 and one edge NE (not

an EE) with target area 2b2. Only one session was explored in Friedman (1996).

Additional data has been collected (Bouchez, 1997) and the results discussed

here are for the combined data. Loose (tight) convergence was found to some BE

in 41 (7) of 46 half-runs, loose (tight) convergence to the EE in 8 (3)
half-runs,

and no loose or tight convergence was found to the edge NE despite its larger

area. The data are sparse but consistent with evolutionary game theory.

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