Stag Hunt • There is no strictly dominant or strictly dominated strategies. • We will find out a player’s best response given the other player’s choice.
Stag Hunt Player 2 Stag Hare Stag Player 1 5, 5 0, 2 Hare 2, 0 2, 2
Stag Hunt Player 2 Stag Stag Player 1 5, 5 Hare 2, 0
Stag Hunt Player 2 Stag Stag Player 1 5, 5 Hare 2, 0
Stag Hunt Player 2 Hare Stag Player 1 0, 2 Hare 2, 2
Stag Hunt Player 2 Hare Stag Player 1 0, 2 Hare 2, 2
Stag Hunt Player 2 Stag Hare Stag Player 1 5, 5 0, 2 Hare 2, 0 2, 2
Stag Hunt Player 2 Stag Hare Stag Player 1 5, 5 0, 2 Hare 2, 0 2, 2
Stag Hunt Player 2 Stag Hare Stag Player 1 5, 5 0, 2 Hare 2, 0 2, 2
Nash Equilibrium (NE) • A pair of strategies form a Nash Equilibrium (NE) if each player is choosing the best response given the other player’s strategy choice. • At a Nash equilibrium, no player can perform a profitable deviation unilaterally.
Strategic Form Game Nash Equilibrium I A Nash equilibrium is such a strategy profile under which no player has the incentive to change his strategy unilaterally. Definition (Pure Strategy Nash Equilibrium) A pure strategy Nash Equilibrium of a strategic form game h I , ( S i ) i 2 I , ( u i ) i 2 I i is a strategy profile s ⇤ 2 S such that for each player i 2 I , the following condition holds u i ( s ⇤ i , s ⇤ � i ) � u i ( s 0 i , s ⇤ 8 s 0 � i ) , i 2 S i . Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 13 / 29
Equilibrium Selection • How to choose between two Nash equilibria? • (Stag, Stag) is payoff dominant : both players get the best payoff possible. • (Hare, Hare) is risk dominant : minimum risk if player is uncertain of each other’s choice. • Many theories, open problem.
Battle of Sexes • A couple need to decide where to go during Friday night. • Husband prefers to go and watch football. • Wife prefers to go and watch ballet. • Both prefer to stay together during the night. • They will make simultaneous decisions during the day without prior communications.
Battle of Sexes Wife Football Ballet Football Husband 4, 2 0, 0 Ballet 0, 0 2, 4
Battle of Sexes Wife Football Football Husband 4, 2 Ballet 0, 0
Battle of Sexes Wife Football Football Husband 4, 2 Ballet 0, 0
Battle of Sexes Wife Ballet Football Husband 0, 0 Ballet 2, 4
Battle of Sexes Wife Ballet Football Husband 0, 0 Ballet 2, 4
Battle of Sexes Wife Football Ballet Football Husband 4, 2 0, 0 Ballet 0, 0 2, 4
Battle of Sexes Wife Football Ballet Football Husband 4, 2 0, 0
Battle of Sexes Wife Football Ballet Football Husband 4, 2 0, 0
Battle of Sexes Wife Football Ballet Husband Ballet 0, 0 2, 4
Battle of Sexes Wife Football Ballet Husband Ballet 0, 0 2, 4
Battle of Sexes Wife Football Ballet Football Husband 4, 2 0, 0 Ballet 0, 0 2, 4
Anti-Coordination Game • In Anti-Coordination Game , it is beneficial for players to choose different strategies.
Hawk-Dove Game • Two birds flight over a valuable territory. • Two possible strategies: • Hawk: flight until injured or your opponent retreats. • Dove: display hostility, but retreat if your opponent chooses to fight.
Hawk-Dove Game Player 2 Hawk Dove Hawk Player 1 -2, -2 2, 0 Dove 0, 2 1, 1
Hawk-Dove Game Player 2 Hawk Hawk Player 1 -2, -2 Dove 0, 2
Hawk-Dove Game Player 2 Hawk Hawk Player 1 -2, -2 Dove 0, 2
Hawk-Dove Game Player 2 Dove Hawk Player 1 2, 0 Dove 1, 1
Hawk-Dove Game Player 2 Dove Hawk Player 1 2, 0 Dove 1, 1
Hawk-Dove Game Player 2 Hawk Dove Hawk Player 1 -2, -2 2, 0 Dove 0, 2 1, 1
Hawk-Dove Game Player 2 Hawk Dove Hawk Player 1 -2, -2 2, 0 Dove 0, 2 1, 1
Hawk-Dove Game Player 2 Hawk Dove Hawk Player 1 -2, -2 2, 0 Dove 0, 2 1, 1
Pure Strategy NE • So far we have restricted each player to choose one strategy. • Also called pure strategy . • The corresponding Nash equilibrium is called pure strategy Nash equilibrium (PNE) .
Mixed Strategies • Sometimes a game may not have a pure strategy Nash equilibrium. • We will allow players to “randomize” over their strategies. • We will see that it can be very natural to do so.
Matching Pennies • Two individuals, each having a penny. • They simultaneously decide to show head or tail of their own penny. • One player prefers to have a matching result. • The other player prefers mismatch.
Matching Pennies Player 2 Head Tail Head Player 1 1, -1 -1, 1 Tail -1, 1 1, -1
Matching Pennies Player 2 Head Tail Head Player 1 1, -1 Tail -1, 1
Matching Pennies Player 2 Tail Head Player 1 -1, 1 Tail 1, -1
Matching Pennies Player 2 Head Tail Head Player 1 1, -1 -1, 1
Matching Pennies Player 2 Head Tail Player 1 Tail -1, 1 1, -1
Matching Pennies Player 2 Head Tail Head Player 1 1, -1 -1, 1 Tail -1, 1 1, -1
Matching Pennies • There is no pure strategy Nash equilibrium. • Given an opponent’s pure strategy, a player always has a profitable deviation: • Player 1 Head -> Player 2 Tail -> Player 1 Tail -> Player 2 Head -> Player 1 Head -> ... • To prevent the opponent from taking advantage, it is better to “randomize” the choices.
Matching Pennies • Assume that player 2 plays Head half of the time and Tail Player 2 half of the time. Head (1/2) Tail (1/2) • Player 1’s expected payoff of choosing Head is: Head 1, -1 -1, 1 Player 1 • 1*(1/2) + (-1) *(1/2) = 0 • Player 1’s expected payoff of -1, 1 1, -1 Tail choosing Tail is: • (-1)*(1/2) + 1 *(1/2) = 0
Matching Pennies • Player 1 is indifferent between Head and Tail, when player 2 chooses according to (1/2, 1/2). • In fact, if player 1 chooses Head with a probability p and Tail with a probability 1- p , then the expected payoff is p *0 + (1- p )*0 = 0. • Hence choosing the two strategies based on ( p , 1- p ) for any p is player 1’s best response.
Matching Pennies • Similarly, assume that player 1 plays Head half of the time Player 2 and Tail half of the time. Head Tail • Player 2’s expected payoff of Head (1/2) choosing Head is: 1, -1 -1, 1 Player 1 • (-1)*(1/2) + 1 *(1/2) = 0 -1, 1 1, -1 Tail (1/2) • Player 2’s expected payoff of choosing Tail is: • 1*(1/2) + (-1) *(1/2) = 0
Matching Pennies • Player 2 is indifferent between Head and Tail, when player 1 chooses according to (1/2, 1/2).
Matching Pennies • (1/2, 1/2) and (1/2, 1/2) are mutual best responses, since no player can find a profitable deviation. • Unique Mixed strategy Nash equilibrium (MNE) : ((1/2, 1/2), (1/2, 1/2)).
Strategic Form Game Mixed Strategy I A mixed strategy is a probability distribution function (or probability mass function) over all pure strategies of a player. I For example, in the Matching Pennies Game, a mixed strategy of player 1 can be σ 1 = (0 . 4 , 0 . 6), which means that player 1 picks “HEADS” with probability 0.4 and “TAILS” with probability 0.6. I Expected Payo ff under Mixed Strategy X Π I � � u i ( σ ) = j =1 σ j ( s j ) · u i ( s ) , s 2 S F σ = ( σ j , ∀ j ∈ I ) is a mixed strategy profile; F s = ( s j , ∀ j ∈ I ) is a pure strategy profile; F σ j ( s j ) is the probability of player j choosing pure strategy s j . Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 16 / 29
Strategic Form Game Mixed Strategy Nash Equilibrium I A mixed strategy Nash equilibrium is such a mixed strategy profile under which no player has the incentive to change his mixed strategy unilaterally. Definition (Mixed Strategy Nash Equilibrium) A mixed strategy profile σ ⇤ is a mixed strategy Nash Equilibrium if for every player i 2 I , u i ( σ ⇤ i , σ ⇤ � i ) � u i ( σ 0 i , σ ⇤ 8 σ 0 � i ) , i 2 Σ i . I In the example of Matching Pennies Game, there is one mixed strategy Nash Equilibrium: σ ⇤ = ( σ ⇤ 1 , σ ⇤ 2 ) with σ ⇤ i = (0 . 5 , 0 . 5), i = 1 , 2. Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 17 / 29
Strategic Form Game “Support” of Mixed Strategy I The “support” of a mixed strategy σ i is the set of pure strategies which are assigned positive probabilities. That is, supp ( σ i ) , { s i 2 S i | σ i ( s i ) > 0 } . Theorem A mixed strategy profile σ ⇤ is a mixed strategy Nash Equilibrium if and only if for every player i 2 I , the following two conditions hold: Every chosen action is equally good, that is, the expected payo ff given σ ⇤ � i of every s i 2 supp ( σ i ) is the same; Every non-chosen action is no better, that is, the expected payo ff given σ ⇤ � i of every s i / 2 supp ( σ i ) must be no larger than the expected payo ff of s i 2 supp ( σ i ) . Huang & Gao ( c � NCEL) Wireless Network Pricing: Chapter 6 October 9, 2018 18 / 29
How to Compute MNE • In general, how do we compute MNE? • In fact, the three games all have MNE. • Stag Hunt • Battle of Sexes • Hawk-Dove • Let’s compute two of them.
Stag Hunt Player 2 Stag Hare Stag Player 1 5, 5 0, 2 Hare 2, 0 2, 2
Stag Hunt Player 2 Stag ( p ) Hare ( 1-p ) Stag ( q ) Player 1 5, 5 0, 2 Hare ( 1-q ) 2, 0 2, 2
Stag Hunt • Given p , Player 1 is indifferent between Stag and Hare. Player 2 • Player 1’s expected payoff of Stag ( p ) Hare ( 1-p ) choosing Stag is 5* p + 0 *( 1-p ). • Player 1’s expected payoff of Stag 5, 5 0, 2 Player 1 choosing Hare is 2* p + 2 *( 1-p ). Hare 2, 0 2, 2 • 5* p + 0 *( 1-p ) = 2* p + 2 *( 1-p ) Hence p = 0.4 • Due to symmetry, q = 0.4.
Unique MNE Player 2 Stag ( 0.4 ) Hare ( 0.6 ) Stag ( 0.4 ) Player 1 5, 5 0, 2 Hare ( 0.6 ) 2, 0 2, 2
Battle of Sexes Wife Football Ballet Football Husband 4, 2 0, 0 Ballet 0, 0 2, 4
Battle of Sexes Wife Football ( p ) Ballet ( 1-p ) Football ( q ) Husband 4, 2 0, 0 Ballet ( 1-q ) 0, 0 2, 4
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