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**Principles of Game Theory**

Lecture 3: Simultaneous Move Games

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**Administrative Problem sets due by 5pm Quiz 1 is Sunday**

Piazza or ~gasper/GT? Quiz 1 is Sunday Beginning or end of class? Questions from last time?

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**Review Simultaneous move situations Backward induction (rollback)**

Strategies vs Actions

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**Normal form games Simultaneous move games Two components to the game**

Many situations mimic situations of 2+ people acting at the same time Even if not exactly, then close enough – any situation where the player cannot condition on the history of play. Referred to as Strategic or Normal form games Two components to the game The strategies available to each player The payoffs to the players “Simple” games often represented as a matrix of payoffs.

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**Cigarette Advertising example**

All US tobacco companies advertised heavily on TV Surgeon General issues official warning Cigarette smoking may be hazardous Cigarette companies fear lawsuits Government may recover healthcare costs Companies strike agreement Carry the warning label and cease TV advertising in exchange for immunity from federal lawsuits. 1964 1970

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**Strategic Interaction: Cigarette Advertising**

Players? Reynolds and Philips Morris Strategies: Advertise or Not Payoffs Companies’ Profits Strategic Landscape Firm i can earn $50M from customers Advertising campaign costs i $20M Advertising takes $30M away from competitor j

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**Strategic Form Representation**

PAYOFFS PLAYERS Philip Morris No Ad Ad Reynolds 50 , 50 STRATEGIES

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**Strategic Form Representation**

PLAYERS Philip Morris No Ad Ad Reynolds 50 , 50 20 , 60 60 , 20 30 , 30 STRATEGIES PAYOFFS PAYOFFS

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What would you suggest? If you were consulting for Reynolds, what would you suggest? Think about best responses to PM If PM advertises? If PM doesn’t? Philip Morris No Ad Ad Reynolds 50 , 50 20 , 60 60 , 20 30 , 30

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**Nash Equilibrium Equilibrium**

Likely outcome of a game when rational strategic agents interact Each player is playing his/her best strategy given the strategy choices of all other players No player has an incentive to change his or her strategy unilaterally Mutual best response. Not necessarily the best outcome for both players.

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Dominance A strategy is (strictly/weakly) dominant if it (strictly/weakly) outperforms all other choices no matter what opposing players do. Strict > Weak ≥ Games with dominant strategies are easy to analyze If you have a dominant strategy, use it. If your opponent has one, expect her to use it.

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**Solving using dominance**

Philip Morris No Ad Ad Reynolds 50 , 50 20 , 60 60 , 20 30 , 30 Both players have a dominant strategy Equilibrium outcome results in lower payoffs for each player Game of the above form is often called the “Prisoners’ Dilemma” Optimal Equilibrium

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**Pricing without Dominant Strategies**

Games with dominant strategies are easy to analyze but rarely are we so lucky. Example: Two cafés (café 1 and café 2) compete over the price of coffee: $2, $4, or $5 Customer base consists of two groups 6000 Tourists: don’t know anything about the city but want coffee 4000 Locals: caffeine addicted but select the cheapest café Cafés offer the same coffee and compete over price Tourists don’t know the price and ½ go to each café

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**Café price competition**

Example scenario: Café 1 charges $4 and café 2 charges $5: Recall: tourists are dumb and don’t know where to go Café 1 gets: 3000 tourists locals = 7K customers * $4 = 28K Café 2 gets 3000 tourists + 0 locals = 3K customers * $5 = 15K Draw out the 3x3 payoff matrix given $2, $4, or $5 price selection (simultaneous selection) 6K tourists and 4K locals.

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**Café price competition**

$2 $4 $5 Café 1 10 , 10 14 , 12 14 , 15 12 , 14 20 , 20 28 , 15 15 , 14 15 , 28 25 , 25 No dominant strategy

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**Dominated Strategies Café 2 $2 $4 $5 Café 1 10 , 10 14 , 12 14 , 15**

A player might not have a dominant strategy but may have a dominated strategy A strategy, s, is dominated if there is some other strategy that always does better than s. Café 2 $2 $4 $5 Café 1 10 , 10 14 , 12 14 , 15 12 , 14 20 , 20 28 , 15 15 , 14 15 , 28 25 , 25

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Dominance solvable If the iterative process of removing dominated strategies results in a unique outcome, then we say that the game is dominance solvable. We can also use weak dominance to “solve” the game, but be careful Player 2 Left Right Player 1 Up 0,0 1,1 Down

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**Weakly Dominated Strategies**

Player 2 Left Right Player 1 Up 0,0 1,1 Down (Down, Right) is an equilibrium profile But so is (Down, Left) and (Up, Right). Why? Recall our notion of equilibrium: No player has an incentive to change his or her strategy unilaterally

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**Fictitious Play Often there are not dominant or dominated strategies.**

In such cases, another method for finding an equilibrium involves iterated “what-if..” fictitious play:

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**Best Response Analysis**

Similarly you can iterate through each strategy and list the best response for the opponent. Then repeat for the other player. Mutual best responses are eq

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Multiple Equilibria We’ve said nothing about there always being a unique equilibrium. Often there isn’t just one:

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**Equilibrium Selection**

With multiple equilibria we face a very difficult problem of selection:

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**Equilibrium Selection**

With multiple equilibria we face a very difficult problem of selection: Imagine Harry had different preferences:

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**Equilibrium Selection**

With multiple equilibria we face a very difficult problem of selection: Classic issues of coordination:

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**No equilibrium in pure strategies**

Nor must there exist an equilibrium in pure strategies Pure strategies means no randomization (penalty kicks) We’ll talk about general existence later Player 2 Rock Paper Scissors Player 1 0,0 -1,1 1,-1

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Multiple players While a X b matrixes work fine for two players (with relatively few strategies – a strategies for player 1 and b strategies for player 2), we can have more than two players: a X b X … X z

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**Homework Study for the quiz**

Next time: more mathematical introduction to simultaneous move games Focus on section 1.2 of Gibbons

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**Equilibrium Illustration**

The Lockhorns:

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Nash equilibrium Nash equilibrium is defined in terms of strategies, not payoffs Every player is best responding simultaneously (everyone optimizes) This.

Nash equilibrium Nash equilibrium is defined in terms of strategies, not payoffs Every player is best responding simultaneously (everyone optimizes) This.

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