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A lot of cooperative game theory deals with transferable utility games, where players can easily transfer "utility" or "payoff" to each other. Another class is non-transferable utility (NTU) games where you get what you get after the strategy profile is implemented and that is that, there are no side transfers. In NTU games it is sensible ...

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then what is the set $\Sigma$ look like? $\Sigma^1$ is the set of all probability distributions over $\{x,y\}$, as such it is given by the one dimensional unit simplex: $$\Sigma^1 = \{(p_1, p_2) \in \mathbb{R}^2_+| p_1 + p_2 = 1\}.$$ Here $p_1$ is the probability of playing $x$ and $p_2$ the probability of playing $y$. If you look at $\Sigma^1$ as a ...

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A strategy profile $\sigma^*=(\sigma_i^*,\sigma_{-i}^*)$ is a Nash equilibrium if for all player $i$, $$u_i(s_i,\sigma_{-i}^*)\ge u_i(s_i',\sigma_{-i}^*), \quad \forall s_i\in\mathrm{supp}(\sigma_i^*), \forall s_i'\in S_i.$$ Rewriting this condition explicitly in terms of the probabilities over pure strategies: \begin{align} \sum_{...

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