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Dynamic Games and Applications
Erschienen in: Dynamic Games and Applications 3/2022

03.03.2022

Cooperation in Dynamic Games with Asymmetric Players: The Role of Social Preferences

verfasst von: Charles F. Mason

Erschienen in: Dynamic Games and Applications | Ausgabe 3/2022

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Abstract

I study a two-player repeated game where payoffs are asymmetric. Attempts to form cooperative agreements in such an environment, for example by pro-rating actions relative to the one-shot Nash equilibrium, have generally struggled in practice. I show that cooperative arrangements when payoff functions are asymmetric tend to disproportionately favor the larger player. Incorporating social concerns, either via envious feelings on the part of the smaller player or altruistic feelings on the part of the larger player, shift quasi-cooperative incentives toward outcomes that are more favorable to the smaller player.
Vorheriger Artikel An Oligopoly-Fringe Model with HARA Preferences
Nächster Artikel Groundwater Exploitation and Illegal Behaviors in a Differential Game

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Fußnoten
1
   United States v. Socony-Vacuum Oil Co., 310 US 150 (1940), reversing 105 F.2d 809 (7th Cir. 1939), reversing 23 F. Supp. 937 (W.D. Wisc. 1938).
 
2
   The Saudis again honored their quota, while Iran produced slightly more than their allocated amount (2454 bpd). By contrast, Venezuela’s substantially exceeded their quota, producing 1852 bpd. See [3, Ch. 4] for discussion.
 
3
   It is not clear that signatories could be forced to remain in any agreement—witness former President Trump’s recent decision to extract the USA from the Paris Agreement; and in any event the actions in the second stage are what really matter.
 
4
   Examples of papers that allow for dynamic interaction include Dockner and van Long [14], Barrett [7], Rubio and Ulph [32], de Zeeuw [12], Mason et al. [29].
 
5
   A second topic for consideration is the nature of the repeated game strategy. While the strategy used in Duffy and Muñoz García [15]—the grim strategy—is often used in the literature, Axelrod [5] suggests a potentially important role for strategies that are more forgiving. To that end, I consider both the grim strategy—under which defection yields perennial “punishment”—and a penal code strategy—which punishes for one period, and then relents in the event that all parties play in accordance with the strategy.
 
6
   A simple condition that ensures this feature of reaction functions is \(\frac{\partial {\uppi }_1(y,y)}{\partial x_1} > \frac{\partial {\uppi }_2(y,y)}{\partial x_2}\).
 
7
   Qualitatively similar results emerge in a repeated game with unknown endpoint, if \({\updelta }\) is interpreted as the probability the game will continue one more period. Alternatively, a collusive regime could emerge if players believe there is a positive probability that their rival is irrational [20, 27].
 
8
   Note that, by construction, \(x_i^N = R_i(x_j^N), i = 1,2.\)
 
9
   Two caveats apply. First, the Nash equilibrium satisfies the pair of conditions Eqs. (1)–(2); as there will be combinations that raise both players’ payoffs so long as \({\updelta } > 0\), I focus on the more cooperative combination. Second, for sufficiently large discount values the players can achieve what we might call “full cooperation,” namely combinations where one player’s payoffs are maximized subject to a minimum payoff constraint for the rival. I suppose the discount factor is too small to engender such an outcome in the discussion below.
 
10
   I note that \(x_i^r< x_i^c < x_i^p\), for both i = 1, 2: the player who is being punished lowers its action while the punishing player raises its action, relative to the carrot action [29, 31].
 
11
   It is algebraically tedious to characterize these various constraints. The code for these numerical analyses, which were conducted in MATLAB, is available upon request.
 
12
   Recall that \(R_1 > R_2\).
 
13
   [35, p. 417] describe a structure where “motivation can be described by a utility function depending on own payoff and deviation from fairness.”
 
14
   The framework Fehr and Schmidt [18] propose is \(U_i({\uppi }_i,{\uppi }_j) = {\uppi }_i - {\upalpha }_i \text {max} ({\uppi }_j - {\uppi }_i, 0) - {\upbeta }_i \text {max} ({\uppi }_i - {\uppi }_j, 0)\) in a two-player setting. With \({\uppi }_1 > {\uppi }_2\), this collapses to \(U_2 = (1 + {\upalpha }_2) {\uppi }_2 - {\upalpha }_2 {\uppi }_1,\) which is equivalent to the description in Eq. (10) when \(\lambda _2 = -{\upalpha }_2/(1 - {\upalpha }_2)\), and \(U_1 = (1 - {\upbeta }_1) {\uppi }_1 + {\upbeta }_1 {\uppi }_2,\) which is equivalent to the description in Eq. (10) when \(\lambda _1 = {\upbeta }_1/(1-{\upbeta }_1).\) Assuming \(1> {\upalpha }_2, {\upbeta }_1 > 0,\) we then have \(\lambda _2< 0 < \lambda _1\).
 
15
   For this diagram, as well as Fig. 5 below, I impose the constraint that player 2 is the small player, i.e., \(s_2 \le 1/2\). While this constraint is irrelevant when \(\lambda _2 = 0\), player 2’s share can exceed 1/2 at small values of \({\upalpha }\) when \(\lambda _2 < 0\).
 
16
   This suggests that envy might potentially offset a considerable part of player 2’s payoff disadvantage.
 
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Metadaten
Titel
Cooperation in Dynamic Games with Asymmetric Players: The Role of Social Preferences
verfasst von
Charles F. Mason
Publikationsdatum
03.03.2022
Verlag
Springer US
Erschienen in
Dynamic Games and Applications / Ausgabe 3/2022
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-022-00435-1

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