On the Existence of Approximate Equilibria and Sharing Rule Solutions in Discontinuous Game

Philippe Bich and Rida Laraki

The notion of non-cooperative games can be a model for numerous conflict situations (such as auction theory, price and quantity competition and vote theory). In this context, the Nash equilibrium is a classic tool for predicting stable agent behaviour: indeed, it describes a situation in which, all things being equal, no agent has an interest in modifying her position (also called a strategy). In a number of economic models, agents exhibit discontinuous preferences: a tiny modification of their strategy can radically change their well-being. One well-known consequence is that it then there may be no stable situation as described by the Nash equilibrium. Two types of responses, generally considered distinct, are given: the first uses the notion of a sharing rule, which fixes the modalities of possible redistribution of agents’ payments, in terms of their strategy.In 1990, Simon and Zame showed that if we authorise the modification of the sharing rule when preferences are discontinuous, then situations of equilibrium reappear, but at a high price, since we must allow agents to adopt random strategies. Whether this same result occurs with non-random strategies remains an open question. In 1999, Reny offered a second type of response: he introduced a new property (“better reply security”), weaker than continuity, in which situations of equilibrium also occur (for random and non-random strategies), but without modifying the sharing rule. Intuitively, this property stipulates that an agent can improve his well-being by changing strategy, but this must be done independently of the infinitesimal changes in strategies made by his competitors.
In this article, Bich and Laraki put together these two approaches and respond positively to the open question: there is a new sharing rule for which there are situations of equilibrium that match non-random strategies. Moreover, if the “better reply security” property is verified, it can be shown that such situations do not, in fact, require modification of the sharing rule: Reny’s and Simon and Zame’s results are thus combined. Modification of the sharing rule can be interpreted as an intervention by an exterior regulatory entity, which stabilises agent behaviour in order to compensate for the fluctuations generated by the discontinuities in their preferences.

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Original title of the article : “On the Existence of Approximate Equilibria and Sharing Rule Solutions in Discontinuous Game”
Published in: Cahier n°2014-22 Ecole Polytechnique, octobre 2014
Available at : https://hal.archives-ouvertes.fr/hal-01071678
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