Games & Strategy 1. Introduction and Representations of a Game (Chapters 2, 3, 4, and 5) 1 Overview • There are various forms of a game ▪ strategic games (also referred to as normal-form games) ▪ extensive-form games ▪ repeated games ▪ Bayesian games 2 Overview • An example of a strategic game: Prisoners’ dilemma Suspect 2 Suspect 1 Quiet Confess Quiet 2, 2 0, 3 Confess 3, 0 1, 1 3 Overview • An example of an extensive-form game 4 Overview • Prisoners’ dilemma can also be played as an extensive-form game ▪ But now it is a different game 5 Overview • An example of a repeated game ▪ Suppose that the Prisoners’ dilemma game is repeated many times ▪ Is there any incentive for players to cooperate? 6 Overview • An example of an incomplete-information game 7 Overview • A strategic game consists of ▪ a set of players ▪ for each player, a set of actions ▪ for each player, preferences over the set of action profiles • See, again, the prisoners’ dilemma Suspect 2 Suspect 1 Quiet Confess Quiet 2, 2 0, 3 Confess 3, 0 1, 1 8 Overview • In the prisoners’ dilemma game ▪ the set of players: {Suspect 1, Suspect 2} ▪ for Suspect 1, the set of actions: {Quiet, Confess} ▪ for Suspect 2, the set of actions: {Quiet, Confess} ▪ Suspect 1’s preference orderings over action profiles: (confess, quiet) > (quiet, quiet) > (confess, confess) > (quiet, confess) ▪ Suspect 2’s preference orderings over action profiles: (quiet, confess) > (quiet, quiet) > (confess, confess) > (confess, quiet) 9 Extensive-form representation • We represent an extensive-form game as a game tree ▪ Nodes represent places where something happens in the game (usually, decisions and termination) ▪ Branches indicate the various actions that players can choose ▪ Such graphical representation is called an extensive-form representation 10 Extensive-form representation • Consider the following extensive-form game 11 Extensive-form representation • In this game, ▪ Players are K (Katzenberg) and E (Eisner) ▪ Each branch indicates their actions ▪ Node a is the initial node ▪ Nodes f, g, h, l, m, and n are terminal nodes (outcomes of the game) ▪ Each player has preference orderings over terminal nodes ▪ Their preferences are represented by payoffs or utilities 12 Extensive-form representation • We use an information set to specify the players’ information at each decision node ▪ Dashed line indicates that the player cannot distinguish between the nodes connected by the dashed line ▪ For example, Katzenberg doesn’t know whether he is at c or d ▪ In other words, he doesn’t know whether Eisner decides to produce “A Bug’s Life” or not. 13 Extensive-form representation • If several nodes are in the same information set, it implies that the player cannot distinguish between the nodes in the information set ▪ For example, when Katzenberg decides whether to leave or stay, he knows he is at a, so information set is {a} ▪ At Eisner’s turn, he knows he is at b (he knows that Katzenberg has left), so information set is {b} ▪ But Katzenberg does not know whether Eisner decided to produce or not, in other words, Katzenberg does not know whether he is at c or d, so information set is {c, d} ▪ After his own decision to produce “Antz”, Katzenberg knows he is at e, so information set is {e} 14 Some terminology • Consider the following price-competition game ▪ We use difference labels (H and H’, L and L’) for player 2’s decision to avoid ambiguity 15 Extensive-form representation • Consider the following price-competition game ▪ Although this game looks a game in which the decisions are made sequentially, it is equivalent to a simultaneous-move game 16 Strategy • A strategy is a complete contingent plan for a player ▪ A “complete contingent” plan means a full specification of a player’s behavior ▪ It describes the actions that the player would take at each of his possible decision points ▪ It describes what the player will do at each of his information sets 17 Strategy • Consider again the following game 18 Strategy • Player 1’s strategy: ▪ play H ▪ play L • Player 2’s strategy: ▪ play H if player 1 plays H and play H’ if player 1 plays L ▪ play H if player 1 plays H and play L’ if player 1 plays L ▪ play L if player 1 plays H and play H’ if player 1 plays L ▪ play L if player 1 plays H and play L’ if player 1 plays L 19 Strategy • Compare with this variant 20 Strategy • Player 1’s strategy is same as before ▪ play H ▪ play L • Player 2’s strategy: ▪ play H ▪ play L 21 Strategy • Strategy space (strategy set) of player i (𝑆𝑖 ) is a set comprising each of the possible strategies of player i ▪ In the previous game, 𝑆1 ={H, L} 𝑆2 ={HH’, HL’, LH’, LL’) ▪ We let 𝑆𝑖 denote a strategy set of player i and 𝑠𝑖 a single strategy for example, 𝑠1 =H, 𝑠2 =HL’ 22 Strategy • A strategy profile is a vector of strategies, one for each player ▪ In the previous case, there are 8 possible strategy profiles ((H,HH’),(H,HL’),(H,LH’),(H,LL’),(L,HH’),(L,HL’),(L,LH’),(L,LL’)) ▪ A typical strategy profile is a vector s=(𝑠1 , 𝑠2 ,…,𝑠𝑛 ) where n is the number of players (i=1,2,….,n) ▪ Let S denote the set of strategy profiles (S=𝑆1 × 𝑆2 × ⋯ × 𝑆𝑛 ) ▪ In the above example, S= 𝑆1 × 𝑆2 ={(H,HH’),(H,HL’),(H,LH’),(H,LL’),(L,HH’),(L,HL’),(L,LH’),(L,LL’)} 23 Strategy • We usually use the term –i to refer everyone except player i ▪ 𝑠−𝑖 is a strategy profile for everyone except player i 𝑠−𝑖 = (𝑠1 , 𝑠2 , … , 𝑠𝑖−1 , 𝑠𝑖+1 , … , 𝑠𝑛 ) ▪ We often simply write 𝑠 = (𝑠𝑖 , 𝑠−𝑖 ) 24 Strategy • In games (a) and (b) (separately), find the strategy sets of each player 25 Normal form • For each player i, we can define a function 𝑢𝑖 : 𝑆 → 𝑹 (𝑹 is the set of real numbers) ▪ In other words, each strategy profile is associated with a real number (payoff or utility) for each player ▪ So for each strategy profile 𝑠 ∈ 𝑆, 𝑢𝑖 𝑠 is player i’s payoff ▪ 𝑢𝑖 is called player i’s payoff function 26 Normal form • Again in the following game ▪ S= 𝑆1 × 𝑆2 ={(H,HH’),(H,HL’),(H,LH’),(H,LL’),(L,HH’),(L,HL’),(L,LH’),(L,LL’)} ▪ for example, 𝑢1 (𝐿, 𝐻𝐻′) = 2, 𝑢2 (𝐻, 𝐿𝐻′) = 2 27 Normal form • In the following game, ▪ 𝑆1 ={OA, OB, IA, IB}, 𝑆2 ={O, I} ▪ S= 𝑆1 × 𝑆2 ={(OA,O),(OA,I),(OB,O),(OB,I),(IA,O),(IA,I),(IB,O),(IB,I)} ▪ For example, 𝑢1 (OA,O)=2, 𝑢1 (IA,I)=4, 𝑢2 (IA,O)=3 28 Normal form • The strategy sets and payoff functions fully describe a strategic situation (without reference to an extensive form) • A game in normal form (or strategic form) consists of ▪ a set of players: {1, 2, …, n} ▪ strategy space for the players: 𝑆1 , 𝑆2 , … , 𝑆𝑛 ▪ payoff functions for the players : 𝑢1 , 𝑢2 , … , 𝑢𝑛 29 Normal form • Two-player normal-form games with finite strategy spaces can be described by matrices (matrix game) • The previous extensive-form game can be represented as a normal-form game 2 1 I O OA 2, 2 2, 2 OB 2, 2 2, 2 IA 4, 2 1, 3 IB 3, 4 1, 3 30 Belief • Belief is a player’s assessment about the strategies of the others in the game ▪ A belief of player i is a probability distribution over the strategies of the other players ▪ Denote such a probability distribution 𝜃−𝑖 ▪ 𝜃−𝑖 ∈ ∆𝑆−𝑖 where ∆𝑆−𝑖 is the set of probability distributions over the strategies of all the players except player i 31 Mixed Strategy • A mixed strategy for a player is the act of selecting a strategy according to a probability distribution ▪ We denote a generic mixed strategy of player i by 𝜎𝑖 ∈ ∆𝑆𝑖 ▪ We call a regular strategy a pure strategy (i.e. chooses one action with 100% probability) 32 Expected Value (Payoff) • When player i has a belief about the strategies of other players, we can calculate the expected value (or expected payoff) of each action in her strategy set ▪ Assume that player i has a belief 𝜃−𝑖 about the strategies of the others and plans to select 𝑠𝑖 ▪ Then expected payoff is 𝑢𝑖 𝑠𝑖 , 𝜃−𝑖 = σ𝑠−𝑖 ∈𝑆−𝑖 𝜃−𝑖 (𝑠−𝑖 )𝑢𝑖 (𝑠𝑖 , 𝑠−𝑖 ) (don’t worry if you don’t understand this mathematical expression) (examples to be shown in class) 33 Rationality • Rationality means maximizing one’s expected payoff ▪ Each player acts to maximize his/her own expected payoff • A player is thought to be rational if ▪ through some cognitive process, he/she forms a belief about the strategies of the others ▪ given this belief, he/she selects a strategy to maximize his/her expected payoff ▪ Nash equilibrium requires additional assumption that players’ beliefs and actual behavior are consistent (more about this later) 34