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Approval mechanism to solve prisoner’s dilemma: comparison with Varian’s compensation mechanism

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Abstract

After having played a prisoner’s dilemma, players can approve or reject the other’s choice of cooperation or defection. If both players approve the other’s choice, the outcome is the result of the chosen strategies in the prisoner’s dilemma; however, if either rejects the other’s choice, the outcome is the same as if they had mutually defected from the prisoner’s dilemma. In theory, such an approval mechanism implements cooperation in backward elimination of weakly dominated strategies, although this is not the case in the subgame perfect Nash equilibrium. By contrast, the compensation mechanism proposed by Varian (Am Econ Rev 84(5):1278–1293, 1994) implements cooperation in the latter but not in the former. This result motivates the present experimental study of the two mechanisms. The approval mechanism sessions yield a cooperation rate of 90% in the first period and 93.2% across periods, while the compensation mechanism sessions yield a cooperation rate of 63.3% in the first period and 75.2% across periods. In addition, the backward elimination of weakly dominated strategies better predicts subjects’ behavior than does the subgame perfect Nash equilibrium in both mechanism sessions.

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Notes

  1. This holds provided the payoff asymmetry in the underlying prisoner’s dilemma is not too large (Qin 2005).

  2. See Hamaguchi et al. (2003), Bracht et al. (2008), and Midler et al. (2015). For Moore-Repullo SPNE mechanism experiments, see Fehr et al. (2014).

  3. Selten (1975) and Kalai (1981) applied BEWDS to game theory.

  4. In this vein, Saijo et al. (2016), who developed a working paper version of the present paper, characterized the AM using certain no-punishment and no-coercion conditions. Kimbrough and Sheremeta (2013) presented situations in which side payments assumed in the CM—collusion in a market, patent races, and R&D competition—might not be legal. Guala (2012) pointed out that there is little anthropological evidence that humankind has used private punishment.

  5. The AM differs from the money-back guarantee mechanism, as follows. If either player, but not both, chooses C, then the $10 contribution is returned to the cooperator. The money-back guarantee mechanism cannot generate (7,17), while the AM achieves (7,17) if players choose (C, D) and then both choose y. The same argument applies to Brams and Kilgour’s (2009) concept of voting between the outcomes of all-C and all-D in order to overcome PD. The AM has an advantage in terms of welfare compared to the money-back guarantee mechanism and Brams and Kilgour’s (2009) voting. To observe this, consider a case in which only player 1 is sufficiently altruistic that player 1’s utility is given by (player 1’s material payoff) \(+{\uprho } \) (player 2’s material payoff), \({\uprho } >4/3\). Assume that players know each other’s payoff function. Then, the outcome of CD maximizes the sum of utilities of two players among four possible outcomes of the PD in Fig. 1. The AM implements CD, while neither the money-back guarantee mechanism nor Brams and Kilgour’s (2009) voting mechanism does.

  6. A path is a list of what each player does in each stage: player 1’s choice between C and D, player 2’s choice between C and D, player 1’s choice between y and n, and player 2’s choice between y and n. For simplicity, hereafter, we write a path as CDny instead of (C, D, n, y). In the AM, we have 96 SPNEs. See Saijo et al. (2016) for more details.

  7. The risk dominance criterion, even with subgame perfection, cannot select the strategy uniquely, unlike in the case of BEWDS. The off-path prediction by risk dominance is redundant. To observe this, consider subgame CD in Fig. 2. According to risk dominance, an NE with greater product deviation loss is more likely to occur (e.g., Blonski and Spagnolo 2015). Since the product of loss deviating from NE (n, y) and that of (n, n) are zero, both are predicted in light of risk dominance. However, seven out of the eight observed choices in subgame CD are (n, y), as reported in Table 2.

  8. These results are due to the discreteness of strategies.

  9. The pairings were anonymous and determined in advance so that no two subjects were paired more than once.

  10. We used Andreoni and Miller’s (1993) method for statistical testing. We first calculated the average cooperation rate for each subject across periods, followed by the test statistic using the averages to eliminate cross-period correlation.

  11. The cooperation rate in the PD is slightly lower than that recorded in previous experiments. For example, Cooper et al. (1996) and Andreoni and Miller (1993) found 20 and 18% cooperation rates, respectively.

  12. Additional experiments on the AM and PD, in which subjects play the game in only one period, provide similar results, suggesting that the AM works without repetition. The data and results are available upon request.

  13. Selten’s index has been used in the literature on testing revealed preference. Our notations follow those of Beatty and Crawford (2011). Forsythe et al. (1996) applied Selten’s index to a voting game experiment.

  14. Although many people advocated the Paris Accord, the national pledges by countries to cut emissions are voluntary and do not involve penalty. “At best, scientists who have analyzed it say it will cut global greenhouse gas emissions by about half the required amount to avert a potential increase in atmospheric temperatures of 2 degrees Celsius or 3.6 degrees Fahrenheit” (NYTimes, 2015).

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Authors and Affiliations

  1. Research Center for Future Design, Kochi University of Technology, 2-2 Eikokuji, Kochi, 780-8515, Japan

    Tatsuyoshi Saijo

  2. Research Institute for Humanity and Nature, 457-4 Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8047, Japan

    Tatsuyoshi Saijo

  3. Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka, 567-0047, Japan

    Takehito Masuda

  4. Tokushima, Japan

    Takafumi Yamakawa

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  1. Tatsuyoshi Saijo
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  2. Takehito Masuda
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  3. Takafumi Yamakawa
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Correspondence to Tatsuyoshi Saijo.

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Additional information

We thank the associate editor and two anonymous referees for their useful comments and suggestions. Masuda is also grateful for helpful comments from Ryan Tierney. This research was supported by the Suntory Foundation; the Joint Usage/Research Center at the Institute of Social and Economic Research, Osaka University; Scientific Research A (24243028) and Challenging Exploratory Research (16K13354) of the Japan Society for the Promotion of Science; the Research Institute for Humanity and Nature (RIHN Project Number 14200122); and “Experimental Social Sciences: Toward Experimentally-based New Social Sciences for the 21st Century,” a project funded by a Grant-in-Aid for Scientific Research on Priority Areas from the Ministry of Education, Science and Culture of Japan. We also thank the Economics Department, Osaka University, for allowing us to use their computer laboratory.

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Saijo, T., Masuda, T. & Yamakawa, T. Approval mechanism to solve prisoner’s dilemma: comparison with Varian’s compensation mechanism. Soc Choice Welf 51, 65–77 (2018). https://doi.org/10.1007/s00355-017-1107-z

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