A recap of notation used in this course

• $\mathbb{R}_{\geq 0}$: used to denote the set of non-negative reals.
• $N$: used to denote number of players and/or number of individuals in a population game.
• $S_i$: set of strategies available to player $i$.
• $u_i:S_i\to \mathbb{R}$: utility function for player $i$.
• $m,n$: used to denote number of strategies.
• $r_i,c_i$: used to denote elements of $S_1$ and $S_2$ in normal form games with $N=2$.
• $\Delta S_i$: the set of mixed strategies of player $i$.
• $\sigma_i$: used to denote an element of $\Delta S_i$.
• $x,y$: used to denote an unkown in a mixed strategy or states in a stochastic game.
• $s^*$: used to denote a best response in a normal form game.
• $UD_i$: used to denote the set of undominated strategies in a normal form game.
• $B_i$: used to denote the set of strategies that are a best response to some strategy.
• $\tau$: used to denote a strategy profile.
• $\tilde s$: used to denote a nash strategy in a normal form game.
• $\mathcal{S}(s)$: used to denote the support of a mixed strategy $\sigma$.
• $T$: used to denote the total number of periods for a repeated game.
• $t$: used to denote the particular repetition in a series of repeated games.
• $U_i$: used to denote the utility to player $i$ in a repeated game.
• $\chi$: used to denote the population vector in a population game.
• $u(s,\chi)$: used to denote the utility of playing $s$ in a population $\chi$.
• $\epsilon$: used to denote proportion of entry population.
• $\chi_{\epsilon}$: used to denote the post entry population.
• $X$: used to denote the set of states for a stochastic game.
• $S_i(x)$: used to denote the set of strategies available to player $i$ in state $x\in X$ in a stochastic game.
• $u(i,x,\tau)$: used to denote the utility to player $i$ in state $x$ given that the strategy profile is $\tau$.
• $\pi(x’|x,\tau)$: used to denote the probability of transferring from $x$ to $x’$ given the strategy profile $\tau$.
• $\lambda$: used to denote the payoff vector for a cooperative game.
• $\pi$: used to denote a permutation.
• $S_{\pi}$: used to denote the set of predecessors of $i$.
• $\phi(G)$: used to denote the Shapley value of a cooperative game $G$.
• $s_i,t_i$: used to denote the sources and sinks of commodity $i$.
• $\mathcal{P}_i$: used to denote the set of paths available to commodity $i$.
• $f^*$: used to denote an optimal flow in a routing game.
• $\tilde f$: used to denote a Nash flow in a routing game.