Abstract
The paper deals with multiplayer normal form games which are preceded by a ‘preplay negotiation phase’ consisting of exchange of preplay offers by players for payments of utility to other players conditional on them playing designated in the offers strategies. The game-theoretic effect of such preplay offers is a transformation of the payoff matrix of the game, obtained by transferring the offered payments between the payoffs of the respective players; thus, certain groups of game matrix transformations naturally emerge. The main result is an explicit and rather transparent algebraic characterization of the possible transformations of the payoff matrix of any given N-person normal form game induced by preplay offers for transfer of payments. That result can be used to describe the ‘bargaining space’ of the game and to determine the mutually optimal game transformations that rational players can achieve by exchange of preplay offers.
Highlights
Some normal form games have no pure strategy Nash equilibria, while others, such as the Prisoners’ Dilemma, have rather unsatisfactory ones, i.e., they are strongly Paretodominated
The key point here is that every preplay offer transforms the given normal form game into another one by an explicitly defined transformation of the payoff matrix
While the proofs are standard applications of linear algebra, I argue that the characterizations themselves are quite useful from game-theoretic perspective as they indicate to the players how they can possibly transform the game they are playing by exchanging suitable preplay offers; describing their “bargaining space“ for that game; 3
Summary
Some normal form games have no pure strategy Nash equilibria, while others, such as the Prisoners’ Dilemma, have rather unsatisfactory ones, i.e., they are strongly Paretodominated. Any preplay offer of player A is binding and irrevocable for A, and it is contingent on B playing the strategy specified by A Such an offer does not create any obligation for B (and, does not transform the game into a cooperative one), as B remains free to choose their strategy when the game is played. By exchanging preplay offers of payments, the players create a preliminary negotiation phase essentially structured as an extensive-form bargaining game. Such scenarios arise in a wide spectrum of economic, social, and political situations, such as collusions, compensations, incentives, concessions, compromises, and other kinds of deals in economic and political negotiations, out-of-court settlements of legal cases, and corruption schemes. I briefly discuss the game-theoretic use and impact of our results
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