# Does Nash Equilibrium can be represented using OWL\SWRL?

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>

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_definition

Received on Monday, 18 November 2013 16:13:45 UTC

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