Introduction to Game Theory (part 1)

1. Introduction to Game Theory(part 1) Chonho Lee Jan 20, 2012

2. What is Game? • Terminology: Players, Actions, Utility, Strategy, Outcome, etc • Classification of game Note that this talk focuses on a strategic game • Strategy • Dominant strategy • Mixed strategy • Solution of game • Strategy profile • Nash equilibrium

3. What is Game Theory • A mathematical formalism for understanding, analyzing, designing and predictingthe outcome of games. • What is a game? • A “structured playing” by Players(2 or more), assumed to be intelligent and rational, that interact with each other by selecting various actions, based on their assigned preferences. • E.g., • Business strategy in economic field • Company consider to open their stores. Which city is the best? • Voting decision in politics • Board game: Chess, Othero, Go, etc. • Network: Routing selection, Resource allocation, etc.

4. What is Game Theory Game consists of • Players (decision makers) • A set of actions available for each player • A set of preference relationships over possible outcome • Strategies • Actions that players choose to follow • Outcome • A set of strategies for all players (or Strategy profile) • It is determined by mutual choice of strategies • Preference relationships • Modeled by utility (or payoff) that users get over outcome

5. Types of Games • Non-cooperative game • Individuals competitions • Cooperative game • Players play as a group including other players • Strategic game • Simultaneous play. Players select their actions at the same time. • Extensive game • Players play or select their actions in order • Repeated game • In dynamic scenario, Players repeatedly play a game • Evolutionary game • In dynamic scenario, Strategy or strategy selection policy vary over time • Complete/Incomplete information game • Information of a game is given/hidden to players

6. Strategic Game • Players select their actions at the same time • G = ( I, {Ai}, U ) • I: a set of players 1, 2, …, N • {Ai}: a set of actions for player i Ai = {a1, a2, …, ak} • U: a set of utilities for players • Denote si as one of actions that player i selects • Utility for player i is represented by ui(s1, s2, …, sN) • Let’s consider • Payoff? • Best response? • Solution (Strategy Profile)? • Nash equilibrium?

7. An Example of Strategic Game • Cold Storage (C) and Fair Price (F) try to open their stores at Boonlay or Pioneer station • Analysis by marketing teams (known information) • 1200 people uses supermarket per day in Boonlay • 300 people uses supermarket per day in Pioneer • Assumption • We assume that ColdStorage attracts twice people than FairPriceif both company open stores at the same station (due to store size) • i.e., • if opened in Boonlay, C gets 800 and F gets 400 people. • If opened in Pioner, C gets 200 and F gets 100 people. • Which station is the best to open stores to get more people??? • Players? Strategies? Utilities?

8. An Example of Strategic Game • We can summarize the game in a table Boonlay (1200) C ? F Pioneer (300)

9. Outcome: Strategy Profile • Outcome of the game? • represented by a set of actions for players  Strategy profile • In our example • 4 possible strategy profiles: (B, B), (B, P), (P, B), (P, P) • For different profiles, players get different payoff • What is the best action for each player?

10. Best Response and Dominant Strategy • Let’s use Payoff table • Best response • a strategy that maximizes payoff given other players’ actions • Focus on Cold Storage • Assume that Fair Price selects Boonlay. Best response? • What if Fair Price selects Pioneer?

11. Best Response and Dominant Strategy • Let’s use Payoff table • Best response • a strategy that maximizes payoff given other players’ actions • Focus on Cold Storage • Assume that Fair Price selects Boonlay. Best response? Boonlay • What if Fair Price selects Pioneer? Boonlay • Regardless of F’s action, C selects Boonlay  Dominant Strategy

12. Best Response and Dominant Strategy • Let’s use Payoff table • Best response • a strategy that maximizes payoff given other players’ actions • Focus on Fair Price • Assume that Cold Storage selects Boonlay. Best response? Boonlay • What if selecting Pioneer? Boonlay • Regardless of C’s action, F selects Boonlay  Dominant Strategy

13. Solution • Best outcome for all players of the game? • A set of best responses to the others’ best response  Solution • In our example, • F’s best response to the best response of C is Boonlay • C’s best response to the best response of F is Boonlay • Solution is a set of dominant strategies for players • If dominant strategies exist for all players

14. Solution • What if one player has no dominant strategy? Cold Storage has a dominant strategy: Boonlay Fair Price does not have a dominant strategy if C selects Boonlay, then F selects Pioneer if C selects Pioneer, then F selects Boonlay

15. Solution • What if one player has no dominant strategy?  How to find the solution?

16. Solution (step by step) • What if one player has no dominant strategy?

17. Solution (step by step) • What if one player has no dominant strategy?

18. Nash Equilibrium • What if one player has no dominant strategy? • Nash Equilibrium [John F. Nash] • A strategy profile that no player can gain higher utility by unilaterally changing his strategy (wiki)

19. Other Examples • What if all players have no dominant strategy?

20. Other Example • What if all players have no dominant strategy? • Which solution is better?

21. Other Examples • What if all players have no dominant strategy? • Which solution is better? • How about this? • No solution???  Extend the concept of strategy

22. Mixed Strategy • A strategy  a probability distribution over possible actions • Assume that A = {a1, a2, …, aN} is a set of actions for all players • A mixed strategy is represented by s = {p(a1), p(a2), …, p(aN)}, • s.t. • p(ak) is a probability that a player selects an action ak • E.g., • let denote si a mixed strategy for player i, and there are 2 possible actions • To obtain the best response, we compute expected utility : A set of actions for players except player i : An k-th action of player j

23. Mixed Strategy • A strategy  a probability distribution over possible actions • Assume that A = {a1, a2, …, aN} is a set of actions for all players • A mixed strategy is represented by s = {p(a1), p(a2), …, p(aN)}, • s.t. • p(ak) is a probability that a player selects an action ak • E.g., • let denote si a mixed strategy for player i, and there are 2 possible actions • To obtain the best response, we compute expected utility

24. Nash Equilibrium by Mixed Strategy • Player 1 selects A1 with probability p and A2 with 1-p • Player 2 selects A1 with probability q and A2 with 1-q • Expected utility? For player 1 • <U1(a1)> = q x 400 + (1-q) x 700 • <U1(a2)> = q x 700 + (1-q) x 500 For player 2 • <U2(a1)> = p x 300 + (1-p) x 400 • <U2(a2)> = p x 200 + (1-p) x 600

25. Nash Equilibrium by Mixed Strategy • Player 1 selects A1 with probability p and A2 with 1-p • Player 2 selects A1 with probability q and A2 with 1-q • Expected utility? For player 1 • <U1(a1)> = q x 400 + (1-q) x 700 • <U1(a2)> = q x 700 + (1-q) x 500 For player 2 • <U2(a1)> = p x 300 + (1-p) x 400 • <U2(a2)> = p x 200 + (1-p) x 600 q <U1(a1)> ? <U1(a2)> 1 2/5 p 0 1

26. Nash Equilibrium by Mixed Strategy • Player 1 selects A1 with probability p and A2 with 1-p • Player 2 selects A1 with probability q and A2 with 1-q • Expected utility? For player 1 • <U1(a1)> = q x 400 + (1-q) x 700 • <U1(a2)> = q x 700 + (1-q) x 500 For player 2 • <U2(a1)> = p x 300 + (1-p) x 400 • <U2(a2)> = p x 200 + (1-p) x 600 q 1 <U2(a1)> ? <U2(a2)> p 0 1 2/3

27. Nash Equilibrium by Mixed Strategy q q q 1 1 1 2/5 2/5 Solution = {s1, s2} = { (2/3, 1/3), (2/5, 3/5) } p p p 0 1 2/3 2/3 0 0 1 1

28. Summary • What is Game? • Terminology: Players, Actions, Utility, etc • Classification of game Note that this talk explains a strategic game • Strategy • Dominant strategy • Mixed strategy • Solution of game • Strategy profile • Nash equilibrium