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.
  • \(r_i,s_i\): 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.