Definitions of noncooperative and cooperative games.

Question

These days I have read many descriptions of a noncooperative game like the one below.

A noncooperative game is a game in which players are unable to make enforceable contracts outside of the rules/description of such a game.

As a graduate student majoring in math, I wonder if there is any mathematically formal definition of a noncooperative( and cooperative) game.

Answer 1

You can define a cooperative game with n players $A=a_1,a_2,\dots a_n$ to be a a function from the power set of $A$ to $\mathbb R$. The idea is each alliance between players has a value, this value can be viewed as a cost or as a reward.

non-cooperative games are usually harder and have many definitions, you might be interested in the extensive way to look at games

Answer 2

Two mathematicians who won Nobel prizes in economics:

1. Nash

2. Shapley

for work in non-cooperative and cooperative game theory respectively. Google and JSTOR are your friends.

Answer 3

I don't think you will ever find a formal definition. The informal one is very good for all purposes. The closest you can get is the definition of a strategic game (= non-cooperative games) see Osborne and Rubinstein (section 2.1) and the definition of coalitional games with and without transferable payoff ( = cooperative games) see again Osborne and Rubinstein (sections 13.1 and 13.5). By the way, Osborne and Rubinstein are freely available online - to get a copy legally, you only need to register.