Decision Making In Prisoner’s Dilemma

6.2 How to study the game (the most important independent variables)

We will now sum up briefly what are the most important variables that are usually studied in the experimental game theoretic research. According to Rapoport (1968) there are two major categories of experimental game theoretic research: a convergent and an divergent design (which happen to be much rarer – it is laborious and even with use of mathematical models it can become computationally unmanageable).

a) In convergent design we study effects of various independent variables on one dependent variable (in Prisoner’s Dilemma studies the dependent variable is, predominantly, the amount of cooperation, operationalized by the frequency or probability of the C choice). Here we give only the most widely studied examples of independent variables, other variables are introduced in section 6.3.

6.21 Effects of learning

The effect of learning is clearly observable after several hundred moves, when subjects tend to lock in on mutual cooperation (CC) or mutual defection (DD), and unilaterally cooperative outcomes are eliminated, see also Pilisuk et al., 1965. “However, the short-term behavior of the subjects is also interesting. In particular, it is interesting to see the distribution of responses in one-shot plays” Rapoport, 1968, p. 465 – these are studies that eliminate the effects of learning.

In Pilisuk et al., 1965 small but significant (p < 0,05) initial differences in cooperativeness (in first five moves) predicted differentiation (in about fifty subsequent moves) of players who locked in on CC (N=52), and those who locked on DD (N=34). We can assume that certain initial conditions lead to certain initial learning that reinforces the initial conditions of relatively higher/lower cooperation.

6.22 Characteristics of subjects

(1) For instance, Hertel et al. (2000) found that people are more likely to cooperate with those in good mood.

(2) Rapoport & Chammah (1965b) found differences in measures of cooperativeness between men and women. When men played against men (MM pairs), they have greater frequency of C than when females played against females (WW pairs). (The study was conducted on 70 male, 70 female and 70 mixed pairs, who played 300 times in succession each.) When men played against women, the amount of cooperation was somewhere in between WW and MM pairs: in WM situation men behaved more like women in WW situation, and women more like men in MM situation. The overall results are summarized in the table below. The greater cooperativeness of male pairs was not due to their greater initial likelihood to cooperate (in the first and second move) which was the same in MM and WW pairs (53%), but by males’ greater ability (propensity) to lock-in on CC exchanges (that is by their greater ability to interact in a mutually beneficial fashion). Also Bedell & Sistrunk (1973) found that males were more cooperative than females in the Prisoner’s Dilemma game, and their findings confirm, too, that females are more cooperative when playing against males than when interacting in same-sex dyads. Caldwell (1976) found no effect of gender on cooperativenes in his five-person Prisoner’s Dilemma experiment.

Table 6.2: Proportion of outcomes in MM, WW, and WM pairs, adapted from Rapoport & Chammah (1965b)

Group	CC	CD	DC	DD
MM	.51	.08	.09	.32
WW	.23	.11	.11	.55
WM	.40	.09	.10	.41

(First letter in outcomes CC, CD, DC, DD refers to the move played by the player represented by the first letter in pairs MM, WW, WM.)

(3) Pilisuk et al. (1965) tested the effect of five personality variables on the amount of cooperation in iterated Prisoner’s Dilemma, namely self-acceptance, monetary risk preference, social risk preference, tolerance for ambiguity, and internationalism (how the variables were defined and assessed, see Pilisuk et al., 1965, p. 494). Only tolerance for ambiguity predicted higher cooperation. Tolerance for ambiguity was measured using a six-point Likert-type scale containing items about preference for regularity, clarity, or ambiguity, balance, or asymmetry. Even more cooperation was to be expected, if not only one but both players of the pair scored high in tolerance for ambiguity.
The best cooperators in the study by Pilisuk et al. (1965) were not the “nicest”, or the least risk-aversive, or the most self-accepting. They were those, who were able to experiment, to reframe the situation (as enabling mutual benefit), and to take appropriate steps that provoked the other player’s cooperation (and then they were able to sustain it). “The ability to reframe old perceptions is a part of tolerance for ambiguity. The finding would seem consistent with the conclusion that a cognitive recasting which occurs during the moves and countermoves of an interpersonal conflict is necessary for the self-organization of the two parties into a single cohesive unit” (Pilisuk et al., 1965, p. 505).
“Tolerance for ambiguity” is possibly not the most exact expression for the cooperation enhancing ability, perhaps it is only a significant part of a bigger set of qualities: 1. seeing the game as a non-zero sum one, 2. seeing the opponent as a potential cooperator, 3. being able to convey (ingeniously, clearly, …) the pro-cooperation message, and 4. having tolerance for some initial costs:
“The Doves [cooperators] as a group were experimenting with cooperation from the start. Some tried and succeeded early, others failed but tried again later and were successful. Some of them use, and some of them report, interesting strategies of rewards and punishments intended to lure the other player toward mutual trust” (Pilisuk et al., 1965, p. 506).
The quality that distinguished cooperators (Doves) from defectors (Hawks) can also be expressed negatively (by what Hawks lacked):
“The most remarkable thing about the Hawks as a group is that they hardly ever tried to cooperate. Following a few unilateral and unreciprocated gestures at the very beginning, the frequency of cooperative gestures (either unilateral or reciprocal) becomes so low that one might conclude that one reason why Hawks never learned to cooperate is because they never experimented with or explored the communication channels that the game was structured to provide. One cannot say of them, as a group, that they are more deceitful or treacherous or aggressively competitive players. On the basis of game performance, what stands out is their conservatism, caution, and reluctance to try a new approach” (Pilisuk et al., 1965, p. 505).
(4) Cooperativeness might be linked to intelligence or various specific cognitive abilities: Jones (2008) in his meta-analysis of iterated Prisoner’s Dilemma games performed at various universities between 1959 and 2003 found that the amount of cooperation increased 5-8% for each 100-point increase in the school’s average SAT score (SAT Reasoning Test, originally Scholastic Aptitude Test measures cognitive skills essential for academic achievement in college, such as mathematics, critical reading, and writing).
(5) Lutzker (1960) and McClintock et al. (1963) found that “isolationists” were more competitive (had lower amount of cooperation) than “internationalists”. Lutzker’s Internationalism-Isolationism Scale was used in the studies:
“[Lutzker] defines an internationalist as one who trusts other nations, is willing to cooperate with them, perceives international agencies such as the United Nations as deterrents of war, and considers international tensions reducible by mediation. The isolationist, on the other hand, is defined as one who demands national strength and might in lieu of international mediation, and does not en courage commerce or transactions with other nations” (McClintock et al., 1963, p. 633)
(6) In a study by Hirsh & Peterson (2009) withdrawal aspect of neuroticism and enthusiasm aspect of extraversion (measured by the Big Five questionnaire) predicted higher cooperation in the Prisoner’s Dilemma.
(7) Terhune (1968), who also offers an overview of Prisoner’s Dilemma research related to personality variables, proposed several reasons for the sometimes somewhat unsatisfactory results of those studies. The most important reason might be this:
“The simple effects of personality were inconsequential compared to their more complex effects, such as the joint effects of two or more personalities in interaction, and the interaction between personality and situational influences; statistically, ‘main effects’ of personality were minor compared to their ‘interaction effects’” (Terhune, 1968, p. 3).
Other reasons include variability of results due to chance factors, irrelevant variables selected for examination, and validity and reliability issues.

6.23 The strategy against which the player plays

We can discern:

(1) Simple contingent strategies. They have a build-in rule how to react to opponent’s (examined subject’s) C and D moves. Such strategies tend to enhance the opponents’ cooperation (Tit for Tat strategy is the most notorious example, see for example Oskamp, 1971; some authors, such as Sermat, 1967, on the other hand, did not find Tit for Tat effective in increasing cooperation in subjects.). Caldwell (1976) importantly noted that Tit for Tat is not “feasible” in variants of the Prisoner’s Dilemma game for three or more players, and that it is difficult, if not impossible to react to opponent’s C or D moves in such variants (when some opponents played C and other opponents D), and hence to “force” an uncooperative player to cooperation is more difficult than in the two-person game (the overall amount of cooperation in an N-person game with conflict of interest was found to be, for example, 46% (Caldwell, 1976) and 32% (Kelly & Grzelak, 1972)).
(2) Non-contingent strategies. That means playing randomly D and C in some given proportion. These strategies have no appreciable effect on cooperation, because the subject cannot see any rule that would be profitable to follow. For example McClintock et al. (1963) found no effect of the competing strategy on his subjects’ decision. Strategies employed in his study responded cooperatively in 85%, 50%, or 15% of trials.
(3) ALL C strategy (always cooperate) leads to bimodal distribution of the subjects, who either play ALL C in response, or exploit the cooperative strategy and play ALL D (always defect).
---
For example Solomon (1960) found that an unconditionally benevolent strategy (ALL C) was exploited by players, while a conditionally benevolent strategy (Tit for Tat: play C on the first move, then reciprocate what the opponent played on the preceding trial) was the best in eliciting cooperation from the opponents.
Deutsch et al. (1967) offered the players a wider variety of moves (1. altruistic behavior: this move will earn money for the other player but not for you, 2. cooperative behavior: this move is worth large amount of money if both players choose it on the same trial, 3. threat (preparing for attack), 4. defensive behavior, 5. aggressive behavior, 6. desire to reform and engage in nonthreatening and nonaggressive activity: you have disarmed, and 7. neutral move: you gain a small reward for yourself, not affecting the other player in any way). Their results are similar to those of Salomon: Turn the Other Cheek strategy (analogous to ALL C) was exploited, while Nonpunitive strategy (analogous to Tit for Tat) was the best in eliciting cooperation. The Deterrent strategy (that, unlike Punitive, attacked back – it even 3. threatened, when the opponent was 4. defensive or 7. neutral) was the worst in eliciting cooperation, and yielded the smallest mean joint payoffs.
After participating in the experiment by Deutsch et al. (1967) subjects rated the strategies on several scales (see Table 6.3 below). Nonpunitive strategy was considered more cooperative, more fair and more stable than Turn the Other Cheek. Turn the Other Cheek is in fact more “cooperative” than Nonpunitive strategy. But the subjects did not perceive the moves of their opponent who played Turn the Other Cheek as more cooperative.
Table 6.3: Subjects’ postexperimental ratings of strategies in Deutsch et al., 1967

	Strategy			Significance level (based on one-way ANOVA, with df for F usually 5/47)
Scale	Turn the Other Cheek	Nonpunitive	Deterrent
Aggressive (1) – Peaceful (9)	7,73	6,60	4,90	p < 0,07
Fair (1) – Unfair (9)	3,09	1,60	3,77	ns
Kind (1) – Cruel (9)	2,30	2,80	4,60	p < 0,1
Stable (1) – Unstable (9)	2,30	1,20	4,60	p < 0,005
Selfish (1) – Generous (9)	7,70	5,60	4,90	p < 0,05
Cooperative (1) – Uncooperative (9)	2,90	1,00	6,20	p < 0,001

6.24 Payoff matrix and incentives

Varying the payoff matrix is how we change one type of game into another. Prisoner’s Dilemma, The Battle of the Sexes, Chicken, and Leader are examples of 2 x 2 games (games with two players with two possible moves each, Rapoport & Guyer, 1966 provide a taxonomical overview). Another widely studied game (not a 2 x 2 game) is Tragedy of the Commons: a renewable resource becomes depleted by excessive harvesting, if the players play D. Interestingly, the Tragedy of the Commons game does not have easily calculated individually rational strategy defined in terms of equilibria; the collectively rational strategy is to claim nothing in the rounds before the final round (the reward doubles in each round) and claim exactly 1/n share in the final round, with n being the number of players, see Rapoport, 1988b.

There are 12 non-equivalent, symmetric payoff matrices for 2 x 2 games. Prisoner’s Dilemma is the only game in this category that has a single so called strongly stable but Pareto-nonoptimal equilibrium – which is to play D; Pareto-optimal move would be to play C, but that strategy is not stable (compare section 6.19).
- In the payoff structure of Prisoner’s Dilemma (where T > R > P > S, 2R > (T + S)) it is understandable that frequency of C increases as R and S increases (these are rewards for cooperators), and conversely frequency of C decreases as T and P increases (these are rewards for defectors), for details see Rapoport (1967b). Axelrod (1967) formulated a measure of conflict (c) for the Prisoner’s Dilemma game (or any other conflict-of-interest game):
c = (T - R) . (T - S) / (T - P)²
The bigger c, the less cooperation is to be expected. As you can see, this measure takes into account all variables in the payoff-matrix (unlike simpler measures proposed by Rapoport & Orwant (1962): c = T - S; or Lave (1965), who suggests c = T - S, c = T - R, or c = P - R). Axelrod’s c has a solid empirical support, since it predicted 0,84 rank-order correlation between c and the percentage of D responses (p < 0,01) (based on extensive data from Rapaport & Chammah, 1965a). Jones et al. (1968) demonstrated (confirmed) linear relation between the frequency (proportion) of cooperation and log (R - P) / (T - S).
- Then there is the question of whether there shall, or shall not be monetary incentives (and how big), and how they affect the players’ attitudes and performance. Oskamp & Kleinke (1970) and Scinto et al. (1972) found no effect of absolute reward increases upon the amount of cooperation in the Prisoner’s Dilemma. Gallo (1966) found that monetary incentives enhance cooperation in experimental games. Gumpert et al. (1969) found that they decrease the amount of cooperation in Prisoner’s Dilemma. Monetary payoffs increased interdyad variability and intradyad uniformity in terms of cooperative behavior (Shaw, 1976).

6.25 Experimental setting

(1) For example: how much time is at disposal for judging and deciding between the alternatives. The more time, it seems, the more likely mutual cooperation is to emerge. If enough time is given, players can begin to understand that they are not necessarily conflicting agents, that they share a non-zero sum game situation. They also have more time to appreciate the logic (intention) of opponent’s moves – they can for example decide to terminate a row of mutual defections, that started as a probing D from player A, was reciprocated by punishing D from player B, which was punished by D from player A, etc. Player A can then show “remorse” and accept a few extra D moves from player B, while playing C, and player B can understand this and take only a little – a symbolic – advantage of this “remorse”: mutual cooperation ensues. Time played some role for example in Pilisuk et al., 1965.

(2) Another variable of experimental setting is the instruction that we give the players, including information about good, optimal, and bad strategies. Such information enhanced cooperation in Rapoport (1988a), in Edney & Harper (1978) it did not; it did partly in Rapoport (1988b) – there was a statistically significant difference (p < 0,1) between informed and uninformed group in the fraction of subjects making no claim on the first round in the Tragedy of the Commons game, the other results (e. g. the difference in mean number of rounds on which first claim was made) were in the expected direction, but not significant.
---
b) In divergent design a set of conditions is hold constant (such as the payoff matrix and subjects) and a great number of dependent variables is examined: such as the conditional frequencies of responses following any of the four possible outcomes CC, CD, DC, DD. We can examine the shifts of propensities (D to C, C to D) following any of the four outcomes (or some patterned sequence of outcomes). We can also study the distribution of the length of runs of various outcomes (or patterned sequences) – we can ask, whether the longer a run (of certain outcome or pattern) lasts, the more likely, or the less likely a shift away from this outcome (pattern) is.