Fuzzy norms, default reasoning and equilibrium selection in games under unforeseen contingencies and incomplete knowledge

This paper concerns the role that norms play in the emergence and selection of equilibrium points seen as social conventions under unforeseen contingencies – that is in the emergence of regularities of behaviour which are self-enforcing and effectively adhered to by limited rational agents due to their self-policing incentives. We define two version of a basic demand game such that in each of them players know that a given solution (norm) is shared knowledge (the Nash product and the typical non cooperative solution of a DP – both coinciding to equilibrium points of the demand game). The basic incomplete knowledge situation is then introduced by assuming that a move of Nature selects states of world where one of the two versions of the basic game will be played. However some of the states of the world that nature may choose are unforeseen. Thus players, when nature has made its choice, will face states that they ex post know to occurs, but that are vaguely described in terms of their ex ante knowledge about the rules of the game. Unforeseen contingencies are then modelled as states that can belong to events defined in terms of the ex ante base of knowledge only through fuzzy membership functions. Players must decide which strategy to play under this characteristic lack of clear information, i.e. under vagueness on the game they are going to play. Their information is resumed by a fuzzy measure of membership that induces a possibility distribution but does not allow them a shared knowledge of the norms (solution), which is played in the given game. Consequently there is no basis at this stage to infer that everybody know that a given equilibrium will be played and to conclude that a player must use his best response belonging to a given equilibrium solution. A default reasoning process here enters the scene, based on the reformulation of default logic in terms of possibility theory given - after Zadeh - by Dubois, Prade and Benferhat. At the first step of the recursive reasoning process, each player's first hypothetical choice, given the basic vague knowledge about the game they are going to play, is calculated in term of a fuzzy expected utility function - where fuzzy utility is considered in conjunction with a possibility measure on unforeseen states. At the second step, each player must guess the simultaneous reasoning process performed by the counterpart. Maybe fallibly, each player attributes by default to the other player his own scheme or reasoning, because it seems to himself the most “normal course of facts” - since he does not know about any falsification of this scheme of reasoning. This is provided by encoding the knowledge base that players have on the game and their default rules of inference - enunciates like “normally players own such information” or “normally a player who owns such information plays strategy such and such in a game like this” - by the formulae of a formal language on which we are able to induce a possibility ordering. The ordering will represent constraints on what the players believe as the “normal course of facts”, which are imposed by the default rules that extends the players' base of knowledge. Then, at the third step we may calculate each player second hypothetical choice given the reconstruction of the other player symmetrical reasoning and the ensuing new possibility assignment on the counterpart's action under any game in each state. This carries to a new best response for each of them in fuzzy expected utility terms. Iterating the procedure will not change the set of predicted choices. So that we can conclude that the default-possibilistic reasoning will stabilize in a couple of strategy choices that constitutes one of the basic games' equilibrium points. We end up by suggesting that the resulting equilibrium will be supported by what in default logic is called the extension of a given theory - which is characterised as a fixed point - obtained by iterately applying to it the set of accepted defaults, without introducing a contradiction. Is to be noted that default logic is non-monotonic, and allows for mistakes and revisions, which seems to belong to the very nature of bounded rationality.