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# Game Theory with Applications to Finance and Marketing

Game Theory with Applications to Finance and Marketing

## where there are two

• Rational players will adopt coalition-proof strategies.In practice, the solution concept of Nash equilibrium is most widelyaccepted. In rare cases only will we turn to other equilibrium concepts.3. Definition 2: A pure strategy is one like U, R, D, or L in example 2. Ithas to be a complete description of a player’s actions taken throughoutthe game. A mixed strategy is a prob. distribution over the set of purestrategies. Immediately, a pure strategy is a mixed strategy.4. Definition 3: Consider a game in normal form,G = (I ⊂ R; {S i ; i ∈ I}; {u i : Π i∈I S i → R; i ∈ I}),where I is the set of players (we are allowing an uncountably infinitenumber of players here), S i is the set of pure strategies feasible toplayer i (also known as the pure strategy space of player i), and u i (·)is player i’s payoff as a function of the strategy profile. If I and S i arefinite for all i ∈ I, then we call G a finite game. We shall representΠ i∈I S i by S (the set of all possible pure strategy profiles). If S i hascardinality m, then the set of feasible mixed strategies for player i,denoted Σ i , is a simplex of dimension m − 1. Each element of Σ i is aprobability distribution over the set S i . We denote a generic elementof S i , Σ i , S, and Σ ≡ Π i∈I Σ i by respectively s i , σ i , s, and σ. Sinceσ i is a probability distribution over S i , we let σ i (s i ) be the probabilityassigned by σ i to the pure strategy s i . Note that being consistent withthe notion of non-cooperative games, the mixed strategies of players areun-correlated. More precisely, given a mixed strategy profile σ, playeri’s payoff isu i (σ) ≡s∈S[Π ∑ I j=1σ j (s j )]u i (s),where note that we have abused the notation a little to let u i (σ) denotethe expected value of u i (s) under the joint probability distribution σ.5. Definition 4: Let σ −i be some element ofΣ −i ≡ Σ 1 × Σ 2 × · · · Σ i−1 × Σ i+1 × · · · Σ I ,where I is the cardinality of I. Let r i (σ −i ) ∈ Σ i be the set of bestresponses of player i against σ −i (i.e., such that u i (r i (σ −i ), σ −i ) ≥3

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