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Introduction to Game theory Presented by: George Fortetsanakis

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Game theory Game theory attempts to mathematically capture behavior in strategic situations, or games, in which an individual's success in making choices depends on the choices of others (Myerson, 1991). A game consists of the following elements – Players: Who participates in the game? – Strategies: What can each player do? – Payoff: What is the outcome of the game?

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Normal form game Consider a simple game between two players α and β. – Player α has n strategies s 1, s 2,..., s n. – Player β has m strategies t 1, t 2, …, t m. Each player receives a payoff when he chooses a certain strategy. – π α (i,j) is the payoff of player α if α chooses the strategy s i and β chooses the strategy t j. – π β (i,j) is the payoff of player β if α chooses the strategy s i and β chooses the strategy t j.

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Bi-matrix representation Let π α and π β be nxm matrices with entries π α (i,j) and π β (i,j). The game in now conveniently represented using the bi-matrix notation.

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Example 1: Oil producing countries 1/2 Two oil producing countries SA and IR can each produce either 2 millions or 4 millions barrels per day. – The total production level will be either 4,6, or 8 millions barrels per day. – Due to market demand the corresponding price per barrel will be $25, S17, or $12. The cost of producing one barrel is $5.

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Example 1: Oil producing countries 2/2 If a player does not know the action of the other player it is preferable to produce 4 million barrels. – Each player will end up earning 28 million dollars. If the two players cooperate they will choose to produce 2 millions barrels each. – Each player will end up earning 40 million dollars.

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Example 2: Rock-Paper-Scissors 1/2 Each player chooses among the strategies s 1 = Rock, s 2 = Paper, or s 3 = Scissors. – Paper wins over Rock, Rock wins over Scissors and Scissors wins over Paper. The winner gets $1 from the loser and no money is exchanged in the case of a tie.

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Example 2: Rock-Paper-Scissors 2/2 Example of Zero sum game: The payoff of one player is negative of the payoff of the other player. Best way to play: Choose any of the three strategies with probability 1/3.

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Mixed strategies The best way to play the Rock, Paper, Scissors game is stochastic and it can be represented with the probability vector: – P = (p R p P p S ) = (1/3, 1/3, 1/3). – This is an example of a mixed strategy. Generalization: We consider a player that can choose among the strategies s 1, s 2, …, s n. We define a mixed strategy as a probability vector upon s 1, s 2, …, s n : – P = (p s1, p s2, …, p sn ), p si ≥ 0,

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Example 3: Hawk and Dove game 1/3 A species of territorial animals engage in fights over territories. Their behavior comes in two variants. – Hawk behavior: Animals fight until either victory or injury ensues. – Dove behavior: Display hostility at first but retreat at the first sign of attack from the opponent. We define: – υ: territory won after a fight. – w: cost of injury.

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Example 3: Hawk and Dove game 2/3 We distinguish the following cases: – Two Hawks meet: If the probability to win is 1/2 the expected payoff of a Hack is υ*1/2 – w*1/2 = (υ-w)/2. – Two Doves meet: Each Dove could win with probability 1/2 thus the expected payoff of a Dove is υ/2. – A Hawk and a Dove meet: The Hawk always wins achieving a payoff υ while the Dove gains nothing.

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Example 3: Hawk and Dove game 3/3 Consider a large population of N animals consisting of N 1 Hawks and N 2 Doves. – When an animal engages in a fight, it meets a Hawk with probability p 1 = N 1 /N and a Dove with probability p 2 = N 2 /N. Expected payoff of a Hawk = Expected payoff of a Dove = The population reaches an equilibrium when p 1 =

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Strategies and payoffs Consider a game in which a player can choose a strategy from the set S = {s 1, s 2, …, s n }. – All members of the set S are called pure strategies. – A mixed strategy is a probability vector on the elements of S. The set of all strategies (pure and mixed) is denoted by Δ(S). – Δ(S) is convex: given two mixed strategies p and q, the convex combination ap + (1-a)q, is also a mixed strategy, ∀ a ∈ [0, 1]

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Example on R 3 If S = {s 1, s 2, s 3 } then Δ(S) is depicted in the following diagram.

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Payoff function A payoff function π : S → R assigns a value π i to each pure strategy s i. We identify the function with the vector: If p is a mixed strategy, the payoff is a random variable whose expected value is the following:

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Best response A strategy s i ∈ S is a pure best response of the payoff π if: A mixed strategy that is a convex combination of pure best response strategies, is also best response for π. Formally a strategy p * is best response for the payoff π if p * maximizes or equivalently:

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Example on R 3 If the pure best response strategies are s 2 and s 3 then the set of all best responses (pure and mixed) are the following:

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Normal form games A finite game in normal form can be described by the following data: A finite set of players Γ = {γ 1, γ 2, … γ n }. A set of pure strategies S γ for each player γ ∈ Γ. – The set S = x γ ∈ Γ S γ is the set of strategy profiles and an element s = (s γ ) γ ∈ Γ assigns a pure strategy to each player. A payoff function π γ : S → R for each player γ ∈ Γ that assigns a payoff to player γ given a strategy profile s.

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Mixed strategy profiles We denote by p γ a (possibly mixed) strategy for the player γ i.e. p γ ∈ Δ(S γ ). The set of mixed strategy profiles is: An element p ∈ Δ contains the mixed strategies that are chosen by all players and is written as p = (p γ1 p γ2, …, p γn ). If p is a mixed strategy profile then the payoff of player γ is a random variable whose expected value is:

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New notation We introduce the notation s -α to denote the pure strategy profile for all players except α, i.e. Similarly we denote by p -α the profile of mixed strategies for all players except player α, i.e.

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Nash equilibrium A Nash equilibrium is a strategy profile p in which no player can improve his payoff by changing his strategy given that the other players leave their own strategy unchanged. Formally, a Nash equilibrium for the game (Γ, S, {π γ } γ ∈ Γ ) is a strategy profile p * ∈ Δ, such that for every γ, p * γ is a best response for the player γ given the strategy profile p -γ of the other players, i.e.

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Example: Matching pennies game 1/4 Two children, holding a penny, independently choose which side of their coin to show. – Child 1 wins if both coins show the same side and child 2 wins otherwise. – The winner pays $1 to the loser.

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Example: Matching pennies game 2/4 Child 1 chooses the mixed strategy p 1 =(p 1,H,p 1,T )= (p, 1-p) Child 2 chooses the mixed strategy p 2 =(p 2,H,p 2,T )= (q, 1-q) Expected payoff for Child 1: Expected payoff for Child 2:

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Example: Matching pennies game 3/4 BR of child 1 to the mixed strategy p 2 of child 2. Solve the above problem using Linear programming BR of child 2 to the mixed strategy p 1 of child 1.

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Example: Matching pennies game 4/4 The NE of the game is the crossing point of BR 1 (p 2 ) and BR 2 (p 1 ). – p 1 =(p 1,H, p 1,T ) = (p, 1-p) and p 2 = (p 2,H, p 2,T ) = (q, 1-q)

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Nash equilibrium Nash equilibrium is defined in terms of strategies, not payoffs Every player is best responding simultaneously (everyone optimizes) This.

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