André Casajus

Outside options, component efficiency, and stability

In this paper, we introduce a component efficient value for TU games with a coalition structure which reflects the outside options of players within the same structural coalition. It is based on the idea that splitting a coalition should affect players who stay together in the same way. We show that for all TU games there is a coalition structure that is stable with respect to this value.

Weakly monotonic solutions for cooperative games

The principle of weak monotonicity for cooperative games states that if a game changes so that the worth of the grand coalition and some players marginal contribution to all coalitions increase or stay the same, then this players payoff should not decrease. We investigate the class of values that satisfy efficiency, symmetry, and weak monotonicity. It turns out that this class coincides with the class of egalitarian Shapley values. Thus, weak monotonicity reflects the nature of the egalitarian Shapley values in the same vein as strong monotonicity reflects the nature of the Shapley value. An egalitarian Shapley value redistributes the Shapley payoffs as follows: First, the Shapley payoffs are taxed proportionally at a fixed rate. Second, the total tax revenue is distributed equally among all players.

Amalgamating players, symmetry, and the Banzhaf value

We suggest new characterizations of the Banzhaf value without the symmetry axiom, which reveal that the characterizations by Lehrer (Int J Game Theory 17:89–99, 1988) and Nowak (Int J Game Theory 26:137–141, 1997) as well as most of the characterizations by Casajus (Theory Decis 71:365–372, 2011b) are redundant. Further, we explore symmetry implications of Lehrer’s 2-efficiency axiom.

The position value is the Myerson value, in a sense

We characterize the position value for TU games with a cooperation structure in terms of the Myerson value of some natural modification of the original game—the link agent form. This construction is extended to TU games with a conference structure.

Monotonic redistribution of performance-based allocations: A case for proportional taxation

Within a simple setup, we show that proportional taxation is implied by three properties: efficiency, symmetry, and monotonicity. Efficiency: redistribution has no cost. Symmetry: members of the society with the same performance obtain the same reward after redistribution. Monotonicity: whenever both the performance of a certain member of the society as well as the overall performance of the society do not decrease, then this members reward after redistribution does not decrease.

The Shapley value without efficiency and additivity

We provide a new characterization of the Shapley value neither using the efficiency axiom nor the additivity axiom. In this characterization, efficiency is replaced by the gain-loss axiom (Einy and Haimanko, 2011, Game Econ Behav 73: 615-621), i.e., whenever the total worth generated does not change, a player can only gain at the expense of another one. Additivity and the equal treatment axiom are substituted by fairness (van den Brink, 2001, Int J Game Theory 30: 309-319) or differential marginality (Casajus, 2011, Theor Decis 71: 163-174), where the latter requires equal productivity differentials of two players to translate into equal payoff differentials. The third axiom of our characterization is the standard dummy player axiom.

Networks and outside options

We introduce and characterize a component efficient value for TU games with a cooperation structure which in contrast to the Myerson (Math Oper Res 2:225–229, 1977) value accounts for outside options. It is based on the idea that the distribution of the worth within a component should be consistent with some connected graph which reflects the outside options of the component’s players.

Weak isomorphism of extensive games

Abstract Based on the Selten [Mathematical Social Sciences 5 (1983) 269] and Oh [Three Essays on Equilibrium Selection in Games, PhD Thesis, (1995) Graduate College of The University of Iowa] symmetries, we introduce weak isomorphism of extensive games that, in contrast to the Peleg et al. [The canonical extensive form of a game form: Symmetries. In: Alkan, A., Aliprantis, C., Yannelis, N. (Eds.), Current Trends in Economics: Theory and Applications, Springer, (1999) pp. 367–387] isomorphism, is compatible with the traditional extensive representation of strategic games. While being sufficiently ‘weak’ to ignore the order of moves to some extent, weak isomorphism is ‘strong’ enough not to violate sequential rationality considerations as incorporated in the concept of sequential equilibrium. In addition, there is some generic equivalence between weak isomorphism and isomorphism of the agent normal form.

Efficient extensions of the Myerson value

We study values for transferable utility games enriched by a communication graph (CO-games) where the graph does not necessarily affect the productivity but can influence the way the players distribute the worth generated by the grand coalition. Thus, we can envisage values that are efficient instead of values that are component efficient. For CO-games with connected graphs, efficiency and component efficiency coincide. In particular, the Myerson value (Myerson in Math Oper Res 2:22–229, 1977) is efficient for such games. Moreover, fairness is characteristic of the Myerson value. We identify the value that is efficient for all CO-games, coincides with the Myerson value for CO-games with connected graphs, and satisfies fairness.

Collusion, quarrel, and the Banzhaf value

We provide new, concise characterizations of the Banzhaf value on a fixed player set employing just the standard dummy player property and one of the collusion properties suggested by Haller (Int J Game Theory 23:261–281, 1994) and Malawski (Int J Game Theory 31:47–67, 2002). Within these characterizations, any of the collusion properties can be replaced by additivity and the quarrel property due to the latter author.