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Game Theory with Applications to Finance and Marketing

Game Theory with Applications to Finance and Marketing

where there are two

where there are two players in the game, who simultaneously chooseactions, and action profiles (U,L), (U,R), (D,L) and (D,R) result inrespectively payoff profiles (0, 1), (−1, 2), (2, −1) and (−2, −2) for thetwo players (where by convention the first coordinate in a payoff profilestands for player 1’s payoff).2. Our purpose of learning the non-cooperatitve game theory is essentiallypractical. In application, we shall first describe an economical orbusiness problem as a game in norm form (or more often in extensiveform; see below), and then proceed to solve the game so as to generateuseful predictions about what the major players involved in the originaleconomical or business problem may do. For this purpose, we need toadopt certain equilibrium concepts or solution concepts. In the remainderof this note we shall review the following solution concepts (andillustrate how those solutions may be obtained by considering a seriesof examples):• Rational players will never adopt strictly dominated strategies;• Common knowledge about each player’s rationality implies thatrational players will never adopt iterated strictly dominated strategies;• Rational players will never adopt weakly dominated strategies;• Common knowledge about each player’s rationality implies thatrational players will never adopt iterated weakly dominated strategies;• Rational players will never adopt strategies that are never bestresponses, or equivalently, rational players will adopt only rationalizablestrategies;• Rational players will adopt Nash equilibrium strategies;• Rational players will adopt trembling-hand perfect equilibrium strategies;• Rational players will adopt subgame-perfect Nash equilibrium strategies;• Rational players will adopt proper equilibrium strategies;• Rational players will adopt strong equilibrium strategies;2

• 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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