Introduction to Game theory

1. Introduction to Game theory Presented by: George Fortetsanakis

2. 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?

3. Normal form game • Consider a simple game between two players α and β. • Player α has n strategies s1, s2, ..., sn. • Player β has m strategies t1, t2, …, tm. • Each player receives a payoff when he chooses a certain strategy. • πα(i,j) is the payoff of player α if α chooses the strategy si and β chooses the strategy tj. • πβ(i,j) is the payoff of player β if α chooses the strategy si and β chooses the strategy tj.

4. Bi-matrix representation • Letπαandπβ be nxm matrices with entries πα(i,j) and πβ(i,j). The game in now conveniently represented using the bi-matrix notation.

5. 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.

6. 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.

7. Example 2: Rock-Paper-Scissors 1/2 • Each player chooses among the strategies s1 = Rock, s2 = Paper, or s3 = 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.

8. 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.

9. Mixed strategies • The best way to play the Rock, Paper, Scissors game is stochastic and it can be represented with the probability vector: • P = (pR pP pS) = (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 s1, s2, …, sn. We define a mixed strategyas a probability vector upon s1, s2, …, sn: • P = (ps1, ps2, … , psn), psi ≥ 0,

10. 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.

11. 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.

12. Example 3: Hawk and Dove game 3/3 • Consider a large population of N animals consisting of N1 Hawks and N2 Doves. • When an animal engages in a fight, it meets a Hawk with probability p1 = N1/N and a Dove with probability p2 = N2/N. • Expected payoff of a Hawk = • Expected payoff of a Dove = • The population reaches an equilibrium when p1 =

13. Strategies and payoffs • Consider a game in which a player can choose a strategy from the set S = {s1, s2, …, sn}. • 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]

14. Example on R3 • If S = {s1, s2, s3} then Δ(S) is depicted in the following diagram.

15. Payoff function • A payoff function π: S → R assigns a value πi to each pure strategy si. 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:

16. Best response • A strategy si ∈ 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 <π, p> or equivalently:

17. Example on R3 • If the pure best response strategies are s2 and s3 then the set of all best responses (pure and mixed) are the following:

18. Normal form games • A finite game in normal form can be described by the following data: • A finite set of players Γ = {γ1, γ2, … γn}. • A set of pure strategies Sγ for each player γ ∈ Γ. • The set S = xγ∈Γ Sγis the set of strategy profilesand an element s = (sγ) γ∈Γassigns a pure strategy to each player. • A payoff function πγ: S → R for each player γ ∈ Γ that assigns a payoff to player γ given a strategy profile s.

19. Mixed strategy profiles • We denote by pγ a (possibly mixed) strategy for the player γ i.e. pγ ∈ Δ(Sγ). 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 = (pγ1pγ2, …, pγn). • If p is a mixed strategy profile then the payoff of player γ is a random variable whose expected value is:

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

21. 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, {πγ}γ ∈Γ)is a strategy profile p*∈ Δ, such that for every γ, p*γis a best response for the player γ given the strategy profile p-γ of the other players, i.e.

22. 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.

23. Example: Matching pennies game 2/4 • Child 1 chooses the mixed strategy p1 =(p1,H,p1,T)= (p, 1-p) • Child 2 chooses the mixed strategy p2 =(p2,H,p2,T)= (q, 1-q) • Expected payoff for Child 1: • Expected payoff for Child 2:

24. Example: Matching pennies game 3/4 • BR of child 1 to the mixed strategy p2 of child 2. • Solve the above problem using Linear programming • BR of child 2 to the mixed strategy p1 of child 1.

25. Example: Matching pennies game 4/4 • The NE of the game is the crossing point of BR1(p2) and BR2(p1) . • p1 =(p1,H , p1,T) = (p , 1-p) and p2 = (p2,H , p2,T) = (q , 1-q)