Introduction to Game theory Presented by: George Fortetsanakis

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?

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.

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.

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.

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.

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.

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.

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,

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.

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.

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 =

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]

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

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:

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:

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:

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.

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:

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.

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.

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.

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:

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.

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)