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.
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.)
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.