Marginality and convexity in partition function form games

In this paper an order on the set of embedded coalitions is studied in detail. This allows us to define new notions of superaddivity and convexity of games in partition function form which are compared to other proposals in the literature. The main results are two characterizations of convexity. The first one uses non-decreasing contributions to coalitions of increasing size and can thus be considered parallel to the classic result for cooperative games without externalities. The second one is based on the standard convexity of associated games without externalities that we define using a partition of the player set. Using the later result, we can conclude that some of the generalizations of the Shapley value to games in partition function form lie within the cores of specific classic games when the original game is convex.

Cartel Formation in Cournot Competition with Asymmetric Costs: A Partition Function Approach

In this paper, we use a partition function form game to analyze cartel formation among firms in Cournot competition. We assume that a firm obtains a certain cost advantage that allows it to produce goods at a lower unit cost. We show that if the level of the cost advantage is “moderate”, then the firm with the cost advantage leads the cartel formation among the firms. Moreover, if the cost advantage is relatively high, then the formed cartel can also be stable in the sense of the core of a partition function form game. We also show that if the technology for the low-cost production can be copied, then the cost advantage may prevent a cartel from splitting.

Computation Problems for Envy Stable Solutions of Allocation Problems with Public Resources

We consider generalizations of TU games with restricted cooperation in partition function form and propose their interpretation as allocation problems with several public resources. Either all resources are goods or all resources are bads. Each resource is distributed between points of its set and permissible coalitions are subsets of the union of these sets. Each permissible coalition estimates each allocation of resources by its gain/loss function, that depends only on the restriction of the allocation on that coalition. A solution concept of "fair" allocation (envy stable solution) was proposed by the author in (Naumova, 2019). This solution is a simplification of the generalized kernel of cooperative games and it generalizes the equal sacrifice solution for claim problems. An allocation belongs to this solution if there do not exist special objections at this allocation between permissible coalitions. For several classes of such problems we describe methods for computation selectors of envy stable solutions.

Cooperative Approaches to Managing Social Responsibility in a Market with Externalities

Problem definition: This paper studies two cooperative approaches of firms in managing social responsibility violations of their supplier: auditing a common supplier jointly (joint auditing) and sharing independent audit results with other firms (audit sharing). We study this problem in a market with externalities and a large number of firms. Academic/practical relevance: With numerous firms procuring their materials and parts worldwide, there are many cases in which overseas suppliers violate safety, labor, or environmental standards. Those violations have externalities in the sense that one firm’s violation affects other firms in the same market. It is not clear how such externalities affect competing firms’ incentives to cooperate and the effectiveness of such cooperation. Methodology: We develop a model based on a cooperative game in partition function form, which enables us to analyze the competitive and cooperative interactions of a large number of firms in a market. Results: Although there has been concern about cooperation for fear of compromising a competitive advantage, firms have incentives to cooperate in managing their suppliers when one firm can be hurt by others’ violations, that is, the negative externality is high. However, neither cooperative approach necessarily improves social responsibility, especially when one firm can benefit from others’ violations, that is, the positive externality is high. Finally, even if agreement is not reached for cooperation before conducting individual audits, social responsibility can still be improved by incentivizing firms to share their private audit results with others under a properly designed mechanism. Managerial implications: The careful assessment of the externalities associated with social responsibility violations is a key to the success of joint auditing and audit sharing. Although firms cooperate voluntarily in some cases, a government agency or an industry association should intervene in other cases to motivate cooperation if it is beneficial. In addition, caution must be taken to monitor manufacturers’ audit efforts, especially when cooperative approaches are implemented in the market where competition is fierce and consumers switch brands easily.

Winning coalitions in plurality voting democracies

We consider plurality voting games being simple games in partition function form such that in every partition there is at least one winning coalition. Such a game is said to be weighted if it is possible to assign weights to the players in such a way that a winning coalition in a partition is always one for which the sum of the weights of its members is maximal over all coalitions in the partition. A plurality game is called decisive if in every partition there is exactly one winning coalition. We show that in general, plurality games need not be weighted, even not when they are decisive. After that, we prove that (i) decisive plurality games with at most four players, (ii) majority games with an arbitrary number of players, and (iii) decisive plurality games that exhibit some kind of symmetry, are weighted. Complete characterizations of the winning coalitions in the corresponding partitions are provided as well.

Linear-State Differential Games in Partition Function Form

We introduce a partition function for [Formula: see text]-player linear-state cooperative differential games. The value of a coalition within a given coalition structure is defined as its noncooperative equilibrium payoff of a game played between the coalitions. We also define two core notions, namely, the cautious and the singleton core. If the game is convex, then the cores are nonempty. In order to illustrate the approach, we consider a symmetric game of pollution accumulation.

The core of a strategic game

In this paper, I introduce and study the $\gamma$-core of a general strategic game. I first show that the $\gamma$-core of an arbitrary strategic game is smaller than the conventional $\alpha$- and $\beta$- cores. I then consider the partition function form of a general strategic game and show that a prominent class of partition function games admit nonempty $\gamma$-cores. Finally, I show that each $\gamma$-core payoff vector (a cooperative solution) can be supported as an equilibrium outcome of an intuitive non-cooperative game and the grand coalition is the unique equilibrium outcome if and only if the $\gamma$-core is non-empty.