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01 Games: introduction

Players, strategies, utilities

Players \(P = \set{p_{i}}\). A strategy is not a move but an algorithmorithm to make moves; compare with decision procedure in decision theory in statistics ref. \(S_{i}\): strategy set of \(p_{i}\). Strategy vector (strategy profile): \(s = (s_{1}, .. s_{n})\). \(s_{-i}\): s sans \(s_{i}\).

Mixed/ randomized strategies

Independent mixed strategy of i: a Prob Distr over \(S_{i}: D_{i}\).

Mixed strategy profile, perhaps \(p_{i}\) coordinated: Probability distribution over \(\times_{i}S_{i}\): D.

Utility

Preference ordering of outcomes for i: Cost, utility of strategy:\ \(c_{i}(s) = -u_{i}(s)\). Compare with risk in decision theory in statistics ref.

eps dominated strategy

\(s_{i}\) is \(\eps\) dominated by \(s_{i}’\) if, \(\forall s_{-i}\) :\ \(u_{i}(s’{i}, s{-i}) \geq u_{i}(s_{i}, s_{-i}) + \eps\). Stronly vs weakly dominated strategy. Incomparable strategies: \(s_{i}\) may be better wrt some \(s_{-i}\), but worse wrt some other \(s_{-i}’\).

Simultaneous move game

Here, strategy = move. Maybe \(\forall i, j, S_{i} = S_{j}\).

Standard (Explicit) form

Cost (or utility) matrix: \(C_{i,j}\): costs (or utility) to P if they select \((s_{i}, s_{j})\). Memory \(\Omega(|S_{i}|^{n})\).

Solution concept

Rule for predicting how game will be played. Prediction is \textbf{solution} or value. Some solution concepts are better than others for certain games.