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Do Small Players Undermine Cooperation in Asymmetric Games?

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Abstract

I study an indefinitely repeated game where players can differ in size and investigate the implications for cooperative-looking behavior. A common approach to modeling such behavior is to appeal to trigger strategies, where deviation from the desired level of cooperation triggers a punishment phase. An alternative approach arises if players condition their actions on a stock of social goodwill, which is developed when players collectively choose more cooperative actions than the one-shot Nash (Benchekroun and Long in J Econ Behav Organ 67(1):239–252, 2008). Using data from two-person experimental games, I analyze an empirical model that combines these two approaches. I find nuanced support for each approach. For the social goodwill model, there is statistically important support in symmetric games and for larger players in asymmetric games, but not for smaller players.

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Notes

  1. A particularly dramatic illustration of this tension occurred in East Texas in the early days of the Great Depression. This situation was complicated by the tension between larger firms, often referred to as majors and smaller firms, often called independents. Libecap and Wiggins [33] show that small producers consistently pumped more than their relative share, measured by acreage or wells in place, and most frequently opposed negotiated efforts to restrict output levels.

  2. In an environment where agents negotiate changes in emissions, asymmetries could make successful negotiation very difficult in the absence of side payments. Incorporating side payments can potentially address this complication, as a number of the papers I discuss below note. That said, I show below that the most cooperative agreement in an asymmetric environment is likely to push the ‘large’ player to indifference between accepting and declining the terms of the agreement, while the ‘small’ player strictly prefers acceptance. In such a regime side payments would need to be paid by the small player to the large player, which seems counter-intuitive (and contrary to the framing of international agreements such as the Clean Development Mechanism).

  3. This earlier paper provides an econometric analysis of paired behavior to compare outcomes in symmetric and asymmetric structures, and finds that outcomes are not significantly different from a Nash equilibrium in the asymmetric treatment. That result left open the question as to whether each individual’s behavior conformed to the Nash equilibrium. My focus in the present paper is on individual player behavior.

  4. For a comprehensive discussion of this literature see [13, 19, 37, 39].

  5. There is a natural isomorphism between the participation game in these models and a merger action in an oligopoly game [30, 48]. Extension of the merger analysis to a multi-period framework is feasible but complex [46].

  6. For an example of the former, see [20]. An example of the latter is [54]; these authors allow for a finite number of moments where coalitions can be reconfigured. Mason et al. [43] provides an example using a differential game, in continuous time.

  7. Extending such political economy based analyses to a dynamic setting is non-trivial; for an example of such an extension, see [44].

  8. Referring to the action following a deviation as a “repentance” action might suggest that playing in a selfish way is somehow sinful. An alternative approach would be to model behavior as embodying “other-regarding preferences” or “ethical choices”. For a survey of such an approach see [38].

  9. Qualitatively similar results emerge in an indefinitely repeated game with unknown endpoint, if \(\delta \) is interpreted as the probability the game will continue one more period.

  10. As such, the strategy combination that chooses these six actions is renegotiation proof [20, 54].

  11. A number of authors have taken this approach, including [12, 18, 32, 36] and [43].

  12. This characterization of average historical contributions can be rewritten as \(Q^a(\tau ) = \left( \frac{\tau - 1}{:}{\tau } \right) Q^a(\tau -1) + \frac{Q^N - Q(\tau )}{\tau }\), which gives a similar flavor to the discrete time characterization for S(t) above. In particular, were \(Q^N - Q(\tau ) = 0\) for multiple consecutive periods \(Q^a(\tau )\) would decline toward zero over time.

  13. This value is reached asymptotically, implying steady state is not reached in finite time. Were an alternative version employed, under which the average past choice is calculated on the basis of a finite number K of past observations, then steady state could be reached in finite time.

  14. Here I am substituting \(\gamma _{1}: = \mu _{1,a} + \nu _{1,a}, \gamma _{2} = \mu _{2,a} + \nu _{1,a}, \gamma _{3} = \mu _{4,a} + \nu _{2,a}\), and \(\gamma _{4} = \mu _{3,a} + \nu _{2,a}\).

  15. I include the experimental instructions that were given to subjects, which includes an example payoff table, in the Appendix.

  16. Relying on all data from an unbalanced panel can generate inconsistent estimates if there is some unobserved explanation for the differing sample periods for different units [55]. Testing for the presence of such an effect can be very difficult, and so I take the conservative approach of using just the first 35 periods in the initial pass through the data. I subsequently analyze the problem using the full data set, for comparison purposes.

  17. See [21] for details. Dynamic stability requires that all of the \(\mu , \nu \) and \(\gamma \) parameters be less than one in magnitude—which they are here. This is a substantive concern, for dynamic stability allows one to interpret the choices derived here as equilibrium choices.

  18. I note that the constant term in this regressions corresponds to the average value of the fixed effects as reported by STATA, the software package I used to obtain the estimates.

  19.    I thank an anonymous referee for suggesting this issue.

  20. Moreover, the departures were roughly similar in magnitude, with large types choosing about 11% below their Nash equilibrium while small types choosing about 7% above their Nash equilibrium—explaining that while the pair’s choice might be close to the combined Nash equilibrium (as found in [41]) the pattern of play departs from a Nash equilibrium in that each type of player’s choice departed from their respective Nash equilibrium value.

  21. See [40] for discussion. One potential explanation for the experimental outcome I describe in the present paper is that small sellers are less patient, i.e.they use a smaller discount factor. While the experimental design that underlies the data I analyze in the current paper is unable to shed light on the role of the discount factor, earlier work does investigate the role f the discount factor [42]. Incorporating such an adjustment to the underlying theoretical framework seems fairly straightforward; I leave such an extension for future research to come to grips with this issue.

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  1. H.A. True Chair in Petroleum and Natural Gas Economics, Department of Economics, University of Wyoming, 1000 E. University Ave., Laramie, WY, 82071, United States of America

    Charles F. Mason

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C.M. executed the empirical analysis, wrote the main manuscript and reviewed the manuscript.

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Correspondence to Charles F. Mason.

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This article is part of the topical collection “Dynamic Games in Economics in Memory of Ngo Van Long” edited by Hassan Benchekroun and Gerhard Sorger.

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Mason, C.F. Do Small Players Undermine Cooperation in Asymmetric Games?. Dyn Games Appl 14, 133–156 (2024). https://doi.org/10.1007/s13235-023-00532-9

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