From: Bai Wei <baiweiyj@gmail.com>

Date: Tue, 19 Nov 2013 00:13:18 +0800

Message-ID: <CA+niQwoZAyzm_kFVts-sBnbTRrL7ymvZ8n8-jD9HBB6OPOd_oQ@mail.gmail.com>

To: semantic-web@w3.org

Date: Tue, 19 Nov 2013 00:13:18 +0800

Message-ID: <CA+niQwoZAyzm_kFVts-sBnbTRrL7ymvZ8n8-jD9HBB6OPOd_oQ@mail.gmail.com>

To: semantic-web@w3.org

Dear all, I want to represent the Nash Equilibrium using OWL\SWRL. I know that the Nash Equilibrium can be represented using First Order Logic. But I do not know whether it can be represented using Semantic Web Languages such as OWL DL, which is based on Description Logic, or SWRL. The informal definition of Nash Equilibrium is [1]: A game (in strategic or normal form) consists of the following three elements: a set of players, a set of actions (or pure-strategies) available to each player, and a payoff (or utility) function for each player. The payoff functions represent each player’s preferences over action profiles, where an action profile is simply a list of actions, one for each player. A pure-strategy Nash equilibrium is an action profile with the property that no single player can obtain a higher payoff by deviating unilaterally from this profile. Formal definition [2]: Let [image: (S, f)] be a game with [image: n] players, where [image: S_i] is the strategy set for player [image: i], [image: S=S_1 \times S_2 \times \dotsb \times S_n] is the set of strategy profiles<http://en.wikipedia.org/wiki/Strategy_(game_theory)> and [image: f=(f_1(x), \dotsc, f_n(x))] is the payoff function for [image: x \in S]. Let [image: x_i] be a strategy profile of player [image: i] and [image: x_{-i}] be a strategy profile of all players except for player [image: i]. When each player [image: i \in \{1, \dotsc, n\}] chooses strategy [image: x_i] resulting in strategy profile [image: x = (x_1, \dotsc, x_n)] then player [image: i]obtains payoff [image: f_i(x)]. Note that the payoff depends on the strategy profile chosen, i.e., on the strategy chosen by player [image: i] as well as the strategies chosen by all the other players. A strategy profile [image: x^* \in S] is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player, that is [image: \forall i,x_i\in S_i : f_i(x^*_{i}, x^*_{-i}) \geq f_i(x_{i},x^*_{-i}).] When the inequality above holds strictly (with > instead of ≥) for all players and all feasible alternative strategies, then the equilibrium is classified as a *strict Nash equilibrium*. If instead, for some player, there is exact equality between [image: x^*_i] and some other strategy in the set [image: S], then the equilibrium is classified as a *weak Nash equilibrium*. *I think the difficult part is how to define the function f that with two arguments. How do you define this function in Semantic Web?* Best Regards, Wei Bai [1] http://www.columbia.edu/~rs328/NashEquilibrium.pdf [2] http://en.wikipedia.org/wiki/Nash_equilibrium#Formal_definitionReceived on Monday, 18 November 2013 16:13:45 UTC

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