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Cooperation in multi-player minimal social situations:
An experimental investigation
Andrew Colman
Briony Pulford
David Omtzigt
Ali al-Nowaihi
An article describing four experiments and a Monte Carlo simulation study carried out for this
project has been published:
Colman, A. M., Pulford, B. D., Omtzigt, D., & al-Nowaihi, A. (2010). Learning to cooperate without
awareness in multiplayer minimal social situations. Cognitive Psychology, 61, 201-227.
A simple, evolutionary game-theoretic model yields the surprising prediction that cooperation can
evolve without deliberate intention in a minimal social situation (MSS). This phenomenon was
discovered in dyads by Sidowski (1957) and generalized to larger groups by Coleman, Colman,
and Thomas (1990). Predictions about the multi-player minimal social situation (MMSS) remain
untested. The primary objective of the proposed research is to test them.
Two-player MSS
In this game of incomplete information, neither play knows the co-player's strategy set nor their
own or the co-player's payoff function. Players may even be ignorant of their strategic
interdependence. Below is the Mutual Fate Control payoff matrix (Thibaut & Kelley, 1959),
normally used to study it:
+, +
+, –
–, +
–, –
All four outcomes are (pure-strategy, weak) Nash equilibria. Players' payoffs are determined by
their co-player's choice and not by their own. Colman (1982, pp. 289-91;1995, pp. 40-50)
suggested some everyday examples.
Experimental evidence
In the earliest experiments (Sidowski, 1957; Sidowski, Wyckoff, & Tabory, 1956), pairs of players
sat in separate rooms with electrodes attached to their left hands. Each player was provided with a
pair of buttons for indicating their choices and a digital display showing points scored. On every
iteration, each player pressed a button, attempting to earn a point and avoid a shock. Pressing
one button awarded the co-player a point, and pressing the other shocked the co-player. In later
experiments, players simply won or lost points – financial incentives have seldom been used.
After many iterations, pairs tended to coordinate on the efficient (C, C) equilibrium. After a
hundred iterations, C-choosing reached 75-80%. Players behaved as if they were learning to
cooperate, although they did not guess that co-players were involved. Sidowski (1957) informed
some players that a person in another room controlled their points and shocks, and vice versa, but
this additional information made no material difference (p. 324). Subsequent investigations of the
MSS, using human and occasionally animal players, have broadly replicated these findings.
MSS theory
To explain the phenomenon, Kelley, Thibaut, Radloff, & Mundy (1962) proposed that players tend
to adopt a myopic “win-stay, lose-change” strategy (Pavlov), repeating a choice immediately
following a reward and switching to the alternative after a punishment. Assume that the game is
repeated indefinitely and that both players then use Pavlov – after arbitrary initial choices. Using 0
and 1 to represent C and D, if both initially choose 0, then both are rewarded and repeat 0
indefinitely: (0, 0) → (0, 0) → (0, 0) → .... If both choose 1, both are punished and switch to 0,
repeating it indefinitely: (1, 1) → (0, 0) → (0, 0) → .... If one player chooses 0 and the other 1, the
0-chooser is punished and switches to 1, and the 1-chooser is rewarded and repeats 1, then both
switch to 0 and repeat it indefinitely: (0, 1) → (1, 1) → (0, 0) → (0, 0) → ... or (1, 0) → (1, 1) → (0,
0) → (0, 0) → ....
Pavlov players therefore lock into mutually rewarding strategies by the third iteration. Pavlov is
essentially a formalization of the law of effect, well documented in human and animal psychology,
but players implement it imperfectly, at best. Strict Pavlov would yield 100% cooperation after
three iterations. Stochastic learning models are more descriptively accurate in the MSS (Arickx &
Van Avermaet, 1981; Delepoulle Preux, & Darcheville, 2000).
Multi-player MMSS
The MMSS is a generalization of the MSS to n ≥ 2 players, each with a uniquely designated
predecessor and successor. Player l's predecessor is Player n and Player n's successor is Player
1, as if the players were seated round a table or the number of players were unbounded. Each
player chooses 0 or 1, yielding an n-vector (configuration) of zeros and ones for each iteration. A
choice of 0 rewards, and a choice of 1 punishes, the successor.
A Pavlov player repeats a rewarded choice on the following iteration and switches after a
punished choice. A jointly cooperative configuration consisting entirely of zeros is repeated
indefinitely. A cooperative configuration is one that leads ultimately to a zero configuration. The
MSS is a special case in which all configurations are cooperative. This is not true in general. The
configuration (1, 1, 0, 0, 1, 1) is followed by (1, 1, 0, 0, 1, 1) → (0, 0, 1, 0, 1, 0) → (0, 0, 1, 1, 1, 1)
→ (1, 0, 1, 0, 0, 0) → (1, 1, 1, 1, 0, 0) → (1, 0, 0, 0, 1, 0) → (1, 1, 0, 0, 1, 1), returning to the
beginning. This sequence cycles forever through these non-cooperative configurations.
Coleman, Colman, and Thomas (1990) proved that, in the MMSS, joint cooperation evolves
only in special cases. The only configurations immediately followed by joint cooperation are those
in which all players choose 1 or all choose 0. If n is odd, then joint cooperation occurs only if all
players make the same initial choice. If k is the highest power of 2 that divides n evenly, then the
number of cooperative configurations is 2k. Once the choices of k players are specified, the rest
are strictly determined for a cooperative configuration.
Proposed experiments
The standard theory yields testable but non-intuitive predictions. For example, cooperation should
not evolve at all in odd-sized groups. However, consider a stochastic modification that we call
Optimistic Pavlov in which a Pavlov player who should choose 1 with certainty chooses it with
probability p (0 < p < 1). Then, immediately following an iteration in which Player i chooses xi and
Player i - 1 chooses xi-1, the probability P(xi) that Player i will defect is P(xi) = p(xi-1 + xi), i = 1, ..., n.
(Here, addition is mod 2, so 1 + 1 = 0, and subscripts mod n, so i - i = 0 = n.) Because 0 < p < 1
and 0 ≤ xi-1 + xi ≤ 1, it follows that P(xi) decreases over iterations. Optimistic Pavlov play therefore
converges towards joint cooperation even in an odd-sized MMSS. We are carrying out
experiments to test these theories, and others, in three-player, four-player, and six-player groups.
Arickx, M., & Van Avermaet, E. (1981). Interdependent learning in a minimal social situation.
Behavioral Science, 26, 229-242.
Coleman, A. A., Colman, A. M., & Thomas, R. M. (1990). Cooperation without awareness: A
multiperson generalization of the minimal social situation. Behavioral Science, 35, 115-121.
Colman, A. M.(Ed.). (1982). Cooperation and competition in humans and animals. Wokingham:
Van Nostrand Reinhold.
Colman, A. M. (1995). Game theory and its applications in the social and biological sciences (2nd
ed.). London: Routledge.
Delepoulle, S., Preux, P. P., & Darcheville, J.-C. (2000). Evolution of cooperation within a
behavior-based perspective: Confronting nature and animats. Artificial Evolution Lecture Notes in
Computer Science, 18291, 204-16.
Kelley, H. H., Thibaut, J. W., Radloff, R., & Mundy, D. (1962) . The development of cooperation in
the “minimal social situation” Psychological Monographs, 76, Whole No. 19.
Sidowski, J. B. (1957). Reward and punishment in the minimal social situation. Journal of
Experimental Psychology, 54, 318-326.
Sidowski, J. B., Wyckoff L. B., & Tabory, L. (1956). The influence of reinforcement and punishment
in a minimal social situation. Journal of Abnormal and Social Psychology, 52, 115-119.
Thibaut, J. W., & Kelley, H. H. (1959). The social psychology of groups. New York: Wiley.
Start Date: 1 October 2004. End Date: 30 June 2005
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