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A game with optimistic aspirations specifies two values for each coalition of players: the first value is the worth that the players in the coalition can guarantee for themselves in the event that they coordinate their actions, and the second value is the amount that the players in the coalition aspire to get under reasonable but very optimistic assumptions about the demands of the players who are not included in the coalition. In this paper, in addition to presenting this model and justifying its relevance, we introduce allocation rules and extend the properties of efficiency, additivity, symmetry, and null player property to this setting. We demonstrate that these four properties are insufficient to find a unique allocation rule and define three properties involving null players and nullifying players that allow the identification of unique allocation rules. The allocation rules we identify are the Midpoint Shapley Value and the Equal Division Rule.

In this paper we introduce games with optimistic aspirations, and we identify two allocation rules for such games—the Midpoint Shapley Value and the Equal Division Rule.

A game with optimistic aspirations specifies two values for each coalition of players: the first value is the worth that the players in the coalition can guarantee for themselves in the event that they coordinate their actions (where the word guarantee implies a very conservative attitude), and the second value is the amount that the players in the coalition aspire to get under reasonable but very optimistic assumptions about the demands of the players who are not included in the coalition.

The two allocation rules that we define on the class of games with optimistic aspirations in this paper, the Midpoint Shapley Value and the Equal Division Rule, are found by extending the axioms that were used in Shapley [

This paper contributes to the field of cooperative game theory. Games with optimistic aspirations are inspired much in the same way in which von Neumann and Morgenstern [

The resulting games with optimistic aspirations incorporate elements of the lower value approach, which associates with each coalition

Games with optimistic aspirations bear similarity to interval games as introduced in Carpente et al. [

Games with optimistic aspirations also bear superficial similarity to games with upper bounds as introduced in Carpente et al. [

Games with optimistic aspirations can be used to shed light on many situations. We name just a few for illustration: in minimal cost spanning tree situations the optimistic values identified in Bergantiños and Vidal-Puga [

The paper is organized as follows. In Section

In this section we recall definitions and results from the literature that we will use later on in this paper.

Let

The class

An allocation rule for TU games is a map that associates a vector

A player

In this section we introduce the definition of games with optimistic aspirations, and we provide the motivation for the introduction of this class of games. Also, we establish that every game with optimistic aspirations can be written as a linear combination of certain basic games with limited aspirations.

A game with optimistic aspirations and set of players

We explain games with optimistic aspirations as well as our motivation for introducing such games by means of a simple example. Consider an interactive situation that can be described by the following 2-player strategic-form game:

Suppose that the two players involved recognize that they can benefit from cooperation and both play their second action (i.e., row 2 and column 2), so that their joint payoffs are 11—much higher than for any other pair of strategies that the players can choose. However, the players have to figure out what side payments would be reasonable to use in order to give both of them the correct incentives to cooperate with each other. One approach to this question is the pessimistic approach that considers the TU-game

A more optimistic perspective is to consider for each coalition

However, as we clearly see in our example, neither the lower value nor the upper value reflects the clear asymmetries that exist between the two players in our simple example. In order to incorporate those, we consider the optimistic aspirations of coalitions—the value that the players in the coalition aspire to get under reasonable but very optimistic assumptions about the demands of the players who are not included in the coalition. We define the optimistic aspirations of coalitions in our example as

We think it is desirable to consider a model that takes this sort of optimistic information into account, while at the same time recognizing that the players have no strategies that guarantee them these optimistic payoffs. To this end, we introduce games with optimistic aspirations, which consist of a TU-game

In our proofs later in this paper, we will use that we can decompose every game with optimistic aspirations

In the following theorem we identify a basis of

Every game with optimistic aspirations can be written as a linear combination of games in the family

Let

If

If

We now turn to demonstrating that the decomposition in (

We can also identify a basis of

Every game with optimistic aspirations can be written as a linear combination of games in the family

Let

The objective of this paper is to find reasonable allocation rules for the class of games with optimistic aspirations

An allocation rule

We have in mind to find an extension of the Shapley Value, and therefore, we start by looking for allocation rules that satisfy axioms similar to those that axiomatize the Shapley Value on the class of games

Two players

A player

In the following example, we demonstrate that the four properties defined previously do not determine a unique allocation rule for games with optimistic aspirations.

We consider convex combinations of the Shapley Values

It is clear from Example

The strong null player property is a direct strengthening of NPP.

The SNPP implies the NPP. To see this, consider a game with optimistic aspirations

We show in the following theorem that the SNPP leads us to the allocation rule

The Midpoint Shapley Value is the allocation rule

The Midpoint Shapley Value

It follows easily from the fact that the Shapley Value satisfies the appropriate efficiency, additivity, symmetry, and null player properties, that

The fact that

Now, let

The Cases 1 and 2 aforementioned demonstrate that

In principle, we can change the weights

Another option one may consider is to take the point of view that if a player

Instead of concentrating on null players, we can also concentrate on nullifying players. Since a nullifying player’s presence prevents others from obtaining a positive worth, the other players may argue that such a player deserves no positive payoff. On the other hand, the nullifying player himself can argue that he deserves no negative payoff either since he can guarantee himself zero by not joining any others. The nullifying player property states that a nullifying player gets a payoff 0. We extend the nullifying player property to the setting of games with optimistic aspirations and investigate if replacing NPP with the new property determines an allocation rule.

A player

We show in the following theorem that the NFPP leads us to the Equal Division allocation rule.

The Equal Division Rule is the allocation rule

The Equal Division Rule

It follows easily and straightforwardly that

Now, let

Remember that

Instead of concentrating on the worths of coalitions that include a nullifying player, we can also concentrate on what happens to the worths of coalitions when a nullifying player joins it. Hence, instead of concentrating on the fact that a nullifying player causes the worth of any coalition he is a member of to be 0, we look at the change in worth that he causes when he joins various coalitions. To reflect this change of focus, we give an alternative (but equivalent) description of nullifying players.

A player

When a nullifying player joins a coalition

It turns out that DPP together with EFF, ADD, and SYM determines a unique allocation rule and that it is the Midpoint Shapley Value

The Midpoint Shapley Value

With regard to existence, it remains to demonstrate that

To prove uniqueness, let

Remember that

In this paper we introduced games with optimistic aspirations in order to be able to capture more of the possible asymmetries between participants in various situations than is possible using existing cooperative game formulations. We also identified two allocation rules for games with optimistic aspirations by first extending the axioms efficiency, additivity, symmetry, and the null player property to the setting of games with optimistic aspirations and, after having shown that the four axioms EFF, ADD, SYM, and NPP do not identify a unique allocation rule, considering three possible alternatives of NPP, namely, the strong null player property, the nullifying player property, and the destroyer player property. We demonstrated that replacing the NPP with the SNPP or DPP leads to the Midpoint Shapley Value, while replacing it with the NFPP leads to the Equal Division Rule.

Thus, we now have a richer way of modeling situations in which there are deeper asymmetries between the participants in a cooperative framework, and we have two methods of allocating the gains from cooperation in such situations that have been identified on the basis of appealing properties. We anticipate that this new methodology will be very useful to obtain new insights into all sorts of cost allocation problems and intend to address specific applications in future research.

The authors thank Julio González-Díaz for suggesting the use of the canonical basis. They also acknowledge the financial support of the University of Santiago de Compostela, of Ministerio de Ciencia e Innovación through Projects ECO2008-03484-C02-02 and MTM2011-27731-C03, and of