222 F.-F. Tang J. of Economic Behavior Org. 44 2001 221–232
players anticipate and react, still adaptively, to the potential instability. When Selten’s anticipation hypothesis is added to Crawford’s dynamic, the mixed equilibrium becomes
stable for small enough adjustment speeds if and only if, a certain stability condition on payoffs is met.
In this paper, Selten’s stability condition, and thus, his anticipations hypothesis, is tested experimentally. Two games are specified. One meets Selten’s stability condition and the
other does not. The games are roughly similar otherwise. For each game, groups of exper- imental subjects are split into two populations to play the game repeatedly under random
matching. Selten’s anticipation hypothesis is supported in the sense that behavior clusters closer to the equilibrium for the game which meets Selten’s stability condition.
Selten’s development is much too long to reproduce. His stability condition can only be sketched. Let ˆ
A be the normalized payoff matrix faced by a player from the first of the two populations. Here “normalized” means that an appropriate constant is added to each column
of the original payoff matrix, say A, to make column sums equal to zero, and an appropriate constant is added to each row of A to make the row sums equal to zero. Similarly, let ˆ
B be the normalized version of the payoff matrix B, faced by a player from the second of the two
populations. Selten’s stability condition is that the non-zero eigenvalues of the matrix ˆ A ˆ
B
T
be negative where T denotes transpose. That is, Selten’s result is this: for small enough adjustment speeds, the mixed equilibrium is stable if and only if the non-zero eigenvalues
of ˆ A ˆ
B
T
are negative. A very rough intuition can be given to the stability condition on ˆ
A ˆ B
T
. Selten gives a much longer and more precise intuition. The effect of the first population’s play on the second
population’s payoff is determined in substantial part by ˆ B
T
, and the effect of the second population’s play on the first population’s payoff is in turn, determined in substantial part by
Â. Thus, the effect of the first population’s play on its own payoffs is critically influenced by the product ˆ
A ˆ B
T
. Under the adjustment dynamic, negative eigenvalues mean, very roughly speaking, that the first population’s change in mixing probabilities are negatively
related to their levels, which creates a stability-promoting effect. From the viewpoint of the second population, a similar interpretation can be given to the transpose of ˆ
A ˆ B
T
. When this stability-promoting effect is present, the instability of the Crawford dynamic is easier
to overcome. It turns out that the effect is necessary and sufficient for Selten’s anticipations mechanism to rescue stability. If the game were constant-sum, then ˆ
B = − ˆ A would hold, and
ˆ A ˆ
B
T
= − ˆ A ˆ
A
T
would automatically have negative eigenvalues also see Conlisk 1993a,b. For the two experimental games presented momentarily, the one failing Selten’s stability
condition will depart further from constant sums. Sections 2 and 3 describe the experimental design and procedures. Section 4 describes
results. Section 5 concludes the paper.
2. Game design
The experiments are based on the following normal form games.For both games, the unique equilibrium is completely mixed with equilibrium probabilities 16, 13, 12 for
both players, and the expected payoff is 10 for both players. However, the non-zero eigen- values of ˆ
A ˆ B
T
are negative for the first game −86.57 and −177.43 and positive for the
F.-F. Tang J. of Economic Behavior Org. 44 2001 221–232 223
second game 12.27 and 91.29. That is, Selten’s stability condition is met for the first game and not for the second.
For game 1, by inspection, player 1 likes diagonal cells and player 2 likes off-diagonal cells somewhat like a poacher-bailiff game, where the three actions are three fish ponds at
which player 1, the poacher, does not want to get caught by player 2, the bailiff. For game 2, there is no clear diagonal versus non-diagonal pattern. Also, the payoff sums, cell by
cell, are somewhat less variable for game 1 than game 2. Nonetheless, it is not clear from inspection why the first game would have greater stability potential than the second. That
is, Selten’s prediction of greater stability for game 1 is not transparent.
Throughout as in the experiment itself, the three actions for player 1 are labelled 1, 2, 3 and for player 2 are labelled 7, 8, 9.
3. Experimental procedure
The experiment was run in the Bonn Experimental Laborotary in 1993. Subjects were recruited by posters from Bonn University; most were law and economics students. No
subject participated in more than one session. Five experimental sessions with 12 subjects in each session were organized for each
game. Each session lasted about 2.5 h one pilot session of 2 h was run 2 weeks before the real sessions. In each session, one game was repeated 150 rounds except the second and
fourth session for game 1, with only 100 rounds, because some subjects spent more time for deliberation during these sessions. Subjects made their decisions at a computer terminal and
did not interact in any other way. Before the experiment started, they were instructed with the aid of projector slides about the game and the use of the computer program. Instructions took
about 12 h and actual play about 2 h. The subjects were also given a one-page reminder of the important information. One protocol, translated into English, is enclosed in Appendix A.
The 12 subjects in a session were randomly divided into two groups of six players each so that they formed two “populations”. The first group took the role of player 1, with pure
strategy choices 1, 2, 3 while the second group took the role of player 2, with pure strategy choices 7, 8, 9. They were matched randomly by computer at each round to play the game.
This set-up was carefully explained to the subjects during the instruction.
The payoffs shown in Fig. 1 are in “points”. On the computer screens, the numbers were in yellow for group 1 and in green for group 2, in blue frames. The exchange rate was
3 Pfennigs per point in the first session for game 1 and 2 Pfennigs per point in the other sessions. One German Mark equals 100 Pfennigs. The exchange rate was always clearly
announced during the instruction period and was typed in bold letters on the reminder page. The instructor also explicitly reminded the subjects that the amount they earned in
the experiment would be paid in cash immediately after the experiment, and that “the more points you get, the more money you get”. The game is easy to understand and the monetary
payment is quite attractive compared to the short time span. On average, subjects made about 18 Marks per hour while the student wage rate in Germany was about 11 Marks per
hour at that time.
At each round, a subject had to choose one and only one strategy. Before confirming a decision, which was highlighted in red on the screens, a subject had the opportunity to change
224 F.-F. Tang J. of Economic Behavior Org. 44 2001 221–232
Fig. 1. The games: payoffs are shown at the upper left corner for player 1 and at the lower right corner for player 2. In the experiment, the strategies are labelled 1, 2, 3 for player 1 and 7, 8, 9 for player 2.
choices. When all 12 subjects made up their mind and sent their choices to the network master, the master program calculated each subject’s payoff according to the matching at
that round and sent this information back to each subject, respectively. Only the subject knew his or her own payoff for a round. Each subject was also informed about: i his or
her opponent’s payoff but not the opponent’s identity, ii the average payoffs and choice frequencies in both groups, iii the “fictitious-play” expected payoffs of the pure strategies
against the other group’s choice distribution and iv his or her total points through that round. A “beep” announced the start of the next round.
Before making decisions, or during the waiting time after having made their decisions, subjects could use the “F1” key to view a summary of previous rounds’ results. They could
choose how many previous rounds’ results to see. For example, if they pressed “F1” at round 20 and chose “5”, they would be shown the summary of results from round 15 to
19. The average payoff and choice frequency of each strategy in both groups, and the “fictitious-play” expected payoffs of the pure strategies against the other group’s choice
distribution, would be shown on the right-hand side of the screen. They could use this
F.-F. Tang J. of Economic Behavior Org. 44 2001 221–232 225
on-line help function as much as they wanted. About half of the subjects never used this function, and only about a fourth of the subjects used this function repeatedly.
All the irrelevant keys on the computer keyboards were “sealed” by the experimenter. The master program recorded vitually everything the subjects had done on their keyboards and
all the information they had seen on their screens. Additionally, each subject was provided with a pencil and one page of blank paper, but only about 20 percent of the subjects ever
used the pencil and paper.
All subjects were interviewed, with the help of the laboratory staff, immediately after the experimental sessions, as they were waiting for their payments in cash. The conversations
were typewritten later by the experimenter. Subjects talked freely about how they made their choices, what they thought during the game playing, what frequency patterns if any they
had observed, how they tried to make use of their observations, and so on. All the detailed records are available from the author upon request.
4. Experimental results
