# ecs289m Spring, 2008 Non-cooperative Games S. Felix Wu Computer Science Departme

## ecs289m Spring, 2008 Non-cooperative Games S. Felix Wu Computer Science Departme

Carlisle, John, Online Editor has reference to this Academic Journal, PHwiki organized this Journal ecs289m Spring, 2008 Non-cooperative Games S. Felix Wu Computer Science Department University of Cali as long as nia, Davis wu@cs.ucdavis.edu http://www.cs.ucdavis.edu/~wu/ Non-cooperative Game Theory The Structure of the Game Four elements Extensive versus Strategic/Normalized as long as ms The Structure of the Game Strategies Mixed, Rationalizable, Dominance Nash Equilibrium Dynamic Game Non-cooperative Game Theory The Structure of the Game Four elements Extensive versus Strategic/Normalized as long as ms The Structure of the Game Strategies Mixed, Rationalizable, Dominance Nash Equilibrium Dynamic Game

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Four Elements A game: Formal representation of a situation of strategic interdependence Set of agents, I (I=n) AKA players Each agent, j, has a set of actions, Aj AKA moves Actions define outcomes For each possible set of actions there is an outcome. Outcomes define payoffs Agents derive utility from different outcomes Matching Pennies Agents: {Alice, Bob} Actions: {Head, Tail} Outcomes: {Matched, Not} Payoffs: {(-1, 1), (1, -1)} Alice gives 1 dollar to Bob! Matching Pennies Agents: {Alice, Bob} Actions: {Head, Tail} Outcomes: {Matched, Not} Payoffs: {(-1, 1), (1, -1)} Bob gives 1 dollar to Alice!

Matching Pennies Simultaneous moves Sequential moves Extensive Form (or Game Tree) Agent Alice Agent Bob H H H T T T Action Terminal node (outcome) Payoffs (-1,+1) (-1,+1) (+1,-1) (+1,-1) Tick-Tack-Toe

Matching Pennies Sequential moves Simultaneous moves Bob doesnt know Alices move Not Perfect In as long as mation Bob can not distinguish Agent Alice Agent Bob H H H T T T Action Terminal node (outcome) Payoffs (-1,+1) (-1,+1) (+1,-1) (+1,-1) Agent Alice Agent Bob H H H T T T Action Terminal node (outcome) Payoffs Another representation (-1,+1) (-1,+1) (+1,-1) (+1,-1)

In as long as mation Sets Agent Alice Agent Bob H H H T T T (-1,+1) (-1,+1) (+1,-1) (+1,-1) Action Terminal node (outcome) Payoffs Matching Pennies Sequential moves Simultaneous moves Bob doesnt know Alices move Not Perfect In as long as mation Assumption: Perfect Recall remember the history Example 1: Perfect Recall p q x x y y a b b a

Example 2: Perfect Recall p q x x y y a b b a b a b a Example 2: Perfect Recall p q x x y y a b b a b a b a Felix likes that as long as sure! Matching Pennies Sequential moves Simultaneous moves Bob doesnt know Alices move Not Perfect In as long as mation Assumption: Perfect Recall Perfect In as long as mation: each in as long as mation set contains a single decision node.

Matching Pennies Sequential moves Simultaneous moves Bob doesnt know Alices move Assumption: Perfect Recall Perfect In as long as mation: each in as long as mation set contains a single decision node. R in addition to om moves Flip a coin to decide Head or Tail Common Knowledge Structure of the Game All the players know about the structure of the game, all players know that their rivals know it, in addition to , Strategy Let HX denote the collection of agent Xs in as long as mation sets, A the set of possible actions in the game, in addition to C(h) A the set of actions possible at in as long as mation set h. A strategy as long as agent X is a function: Per player (Info Set => Action)

Strategies as long as Matching Pennies Bobs four possible strategies: S1: Play H if Alice plays H; play H if Alice plays T S2: Play H if Alice plays H; play T if Alice plays T S3: Play T if Alice plays H; play H if Alice plays T S4: Play T if Alice plays H; play T if Alice plays T In as long as mation Sets Strategies imply A sequence of moves actually taken A probability distribution over the terminal nodes of the game Strategies ~ Outcomes ~ Payoffs Strategies imply A sequence of moves actually taken A probability distribution over the terminal nodes of the game Strategies ~ Outcomes ~ Payoffs Strategic/Normal Forms

Matching Pennies Alice Bob H H T T -1, +1 -1, +1 +1, -1 +1, -1 Action Outcome Payoffs Bob doesnt know Alices move Alice Bob H H T T -1, +1 -1, +1 +1, -1 +1, -1 Action Outcome Payoffs Strategies Strategies as long as Matching Pennies Bobs four possible strategies: S1: Play H if Alice plays H; play H if Alice plays T S2: Play H if Alice plays H; play T if Alice plays T S3: Play T if Alice plays H; play H if Alice plays T S4: Play T if Alice plays H; play T if Alice plays T In as long as mation Sets

Bob does know Alices move Alice Bob H S1 T -1, +1 +1, -1 +1, -1 S2 S3 S4 -1, +1 -1, +1 +1, -1 +1, -1 -1, +1 Alice Bob H S1 T -1, +1 +1, -1 +1, -1 S2 S3 S4 -1, +1 -1, +1 +1, -1 +1, -1 -1, +1 S2: Play H if Alice plays H; play T if Alice plays T Non-cooperative Game Theory The Structure of the Game Four elements Extensive versus Strategic/Normalized as long as ms The Structure of the Game Strategies Mixed, Rationalizable, Dominance Nash Equilibrium Dynamic Game

Whats next Many mathematical tools, definitions, in addition to theories Now, lets try to apply them to online social networks Materials Mainly Covered Chapter 5 Chapters 7~9

## Carlisle, John Online Editor

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