“Greedy” demand adjustment in cooperative games

Annals of Operations Research https://doi.org/10.1007/s10479-023-05179-8 ORIGINAL RESEARCH “Greedy” demand adjustment in cooperative games Maria Montero1 · Alex Possajennikov1,2 Accepted: 6 January 2023 © The Author(s) 2023 Abstract This paper studies a simple process of demand adjustment in cooperative games. In the process, a randomly chosen player makes the highest possible demand subject to the demands of other coalition members being satisfied. This process converges to the aspiration set; in convex games, this implies convergence to the core. We further introduce perturbations into the process, where players sometimes make a higher demand than feasible. These perturbations make the set of separating aspirations, i.e., demand vectors in which no player is indispensable in order for other players to achieve their demands, the one most resistant to mutations. We fully analyze this process for 3-player games. We further look at weighted majority games with two types of players. In these games, if the coalition of all small players is winning, the process converges to the unique separating aspiration; otherwise, there are many separating aspirations and the process reaches a neighbourhood of a separating aspiration. Keywords Demand adjustment · Aspirations · Core · Stochastic stability 1 Introduction In transferable utility cooperative games, we consider the following process. Suppose players currently have some demands. These demands can be interpreted as what they expect from the game. A player is randomly selected. This player is in a position to propose a coalition; but the coalition partners agree to form it only if their demands are satisfied. The player looks for a coalition that, after the demands of the coalition partners are subtracted from the coalition’s worth, leaves the most to the player. The player then makes the demand equal to the residual. The payoff vectors that allow each player to achieve such “maximal” demands in at least one coalition are called aspirations in Bennett (1983). (They are also called semi-stable demand vectors in Albers, 1979; Selten, 1981). Bennett et al. (1997) show that the process described in the previous paragraph converges to the set of aspiration payoff vectors. B Alex Possajennikov 1 University of Nottingham, Nottingham, UK 2 School of Economics, University of Nottingham, Nottingham NG7 2RD, UK 123 Annals of Operations Research We analyze the implications of the process further. First, we show that in convex games the Bennett et al. (1997) result implies that the demand adjustment process converges to the core of the game.1 The set of aspirations is in general quite large, and there can be aspirations where some players demand very little. Cross (1967) argues that “scarce” players (players that are underdemanding and hence sought after as coalition partners) should be able to increase their demands. We formalize this argument by adding to the process the possibility of “mutations”. With a small probability a player makes a demand different from the maximal feasible one; instead, the player (most likely) makes a higher demand. Since in the basic process players make the maximum feasible demands and the most likely mutations are to even higher demands, the process overall can be seen as “greedy”. We show that separating aspirations (a subset of partnered aspirations, defined in Albers, 1979; Bennett, 1983), i.e., demand vectors in which no player is indispensable in order for other players to achieve their demands, are the ones most resistant to such upward mutations. We fully analyze the process with mutations in 3-player superadditive games. In these games, if the core is non-empty, demand vectors that are in the core are stochastically stable (meaning that, as the mutation probability goes to zero, the process spends almost all of the time in the core). If the core is empty, the unique separating aspiration is stochastically stable. We then turn to weighted majority games.2 In Montero and Possajennikov (2022), we showed that separating aspirations are stochastically stable in symmetric weighted majority games and in apex games. In this paper, we analyze weighted majority games with two types of players further.3 In these games, if there are enough small players (i.e., if the coalition of all small players is winning), the process converges to the unique separating aspiration. On the other hand, if the coalition of all small players is losing, then there are many separating aspirations and the process reaches a neighborhood of a separating aspiration. The paper contributes to the literature, reviewed in Newton (2018, Sect. 6), that applies evolutionary approaches to predicting outcomes in cooperative games. Agastya (1997) has a demand adjustment process in which players simultaneously make demands, and a coalition compatible with demands forms (with some probability, if several coalition structures are compatible). Using a myopic best response to incomplete memory samples, Agastya shows that in convex games the process converges to the core. Rozen (2013) allows the players, in addition to demands, to also name a list of potential coalition partners, obtaining the same result. With our process (without mutations), convergence to the core in convex games follows from the observation that in convex games the set of aspirations coincides with the core (Moldovanu & Winter, 1994). As in Agastya (1997) and Rozen (2013), in our process the players need to know the (previous) demands of other players (and the characteristic function of the game) in order to find the best demand to make. In the demand adjustment process in Nax (2010, chapter 4), a player can increase or decrease the demand (although only by a small amount), depending on whether the player is in a coalition that satisfies the demand but without knowing the demands of others. The process predicts outcomes close to the core (but not exactly in the core, since players can temporarily increase demands with a non-zero probability) if the core 1 Agastya (1997) and Rozen (2013) have this result for similar adjustment processes. We discuss the differences between their and our models in the discussion of related literature later in the introduction. 2 (Weighted) majority games are a class of games in which some coalitions can “win” (have a positive worth) while others “lose” (have zero worth). Such games are often studied in political science and economics in the context of voting (for example, Felsenthal and Machover (1998), and, more recently, Kurz et al. (2023)). 3 These games are often found in practice. For example, the distribution of party seats in the current German parliament (Bundestag) gives rise to such a game (see Example 2 in Sect. 5.2). 123 Annals of Operations Research is non-empty. Using a variant of Nax’s process, Issleib (2015) shows that in 3-player games an equitable allocation in the core is selected; this is related to the assum (...truncated)