A Non-Cooperative Shapley Value Representation of Luce Contests Success Functions

In the literature on conflicts, the rule that determines the 'win probabilities' of the contestants is often specified by a mapping –contest success function (CSF)— which translates the effort vectors into a set of win probabilities. When these mappings correspond to the choice probabilities of the Luce model (Luce (1959)), this gives rise to the popular Luce-contest success function (CSF). The use of this specific model in conflicts remains unclear: How does the Luce CSF rule get generated in the first place and what are the choice probabilities actually representing in a conflict? Are they individual probabilities of winning the prize, or a share of the resources allocated to each contestant? This paper shows that the Luce CSF can take on these two interpretations simultaneously within a single non-cooperative environment. We carry out this exercise by following and extending the strategic approach of the Shapley value initiated by Ui (2000). Our main methodological innovation is to connect the class of TU games with action choices of a strategic game to its aggregate deviation functions. Considering a class of anti-coordination games, we then obtain two main results. Our firstmain theoremstates that the Luce values –which represent the 'impact functions' in the case of a contest– are given by the Shapley value of the TU games with action choices associated to the non-cooperative game when the players' belief are in equilibrium. Our second 'representation theorem' relates the axioms given by Skaperdas (1996) to represent the logit CSF as the solution of the TU-games associated to the non-cooperative game when the players' belief are in equilibrium and the axioms of Shapley hold for this solution. In this case, our approach singles out the specific class of Luce CSFs of the Tullock and power-forms as the only possible forms of CSFs. Hence, in this sense, our results show that the Luce CSFs can be given a 'non-cooperative Shapley' representation. As a corollary, we discuss how our approach may also provide a non-cooperative theory to the quasivalue order representation of stochastic rules introduced in Monderer and Gilboa (1992).