Introduction to Game Theory: Basics, Dominance, Nash ...shihenl/302/4p Normal-Form Games.pdf ·...

Introduction to Game Theory: Basics, Dominance, NashEquilibrium

Economics 302 - Microeconomic Theory II: Strategic Behavior

Shih En Lu

Simon Fraser University(with thanks to Anke Kessler)

ECON 302 (SFU) Introduction to Game Theory 1 / 31

Topics

1 Game theory: basic definitions2 (Strict) Dominance3 Pure-strategy Nash equilibrium4 Mixed-strategy Nash equilibrium5 Best response graphs in 2x2 games6 Mixed strategies and Dominance


Most Important Things to Learn

1 Definitions: strategy, strategy profile, dominant/dominated strategy,iterated strict dominance (ISD), best response, Nash equilibrium (NE)

2 Determine the set of strategies that survive ISD3 Find a game’s pure-strategy NE4 Finding mixed-strategy NE in 2x2 games (or games that become soafter ISD)

5 Graphing best response functions in 2x2 games


Why Games?

We want to model strategic behaviour - i.e. how people act whenthey realize that their own behaviour affects others’choices and thatthe outcome depends on others’actions as well as their own.

This fits many economic situations where the "large number"assumption of competitive markets fail: oligopoly, auctions,bargaining, public goods, etc.

Also useful in other fields: biology, political science, sports, etc.


What is a Game?

A game has players that each chooses from actions available tohim/her.

Example: Bob can either go to class or skip class, and the professorcan either give a pop quiz or not.

This week, we study simultaneous-move games.

Simultaneous-move means that each player acts once, and does sowithout knowing others’actions/strategies.

Example: The game would not be simultaneous-move if the professordecides whether to give a quiz after seeing if Bob is in class.


What is a Strategy (in a Simultaneous-Move Game)?

Each player plays a strategy, which is, in a simultaneous-move game,a probability distribution over her actions.

Example: "Go to class" is one of Bob’s strategies. "Go to class withprobability 0.3 and Skip class with probability 0.7" is another.

A pure strategy is a strategy that puts probability 1 on a single action.A mixed strategy is just any strategy. "Mixed" is used to emphasizethat the strategy may not be pure.

A collection of each player’s strategy is called a strategy profile.Example: ((Go to class), (0.4 Give quiz, 0.6 No quiz)) is a strategyprofile.


Outcomes, Payoffs and Complete Information

An outcome (or action profile) is a collection of actions taken byeach player.

Example: (Go to class, Give quiz) is an outcome.

Each outcome generates a utility, or payoff, for each player. Theseutilities are obtained from expected utility theory, so that a playercares about her expected payoff/utility.Complete information means that each player knows every player’spayoff from each outcome.

Example: If lazy students’payoffs differ from hard-working ones’, andthe professor does not know whether Bob is lazy, then the gamewould not feature complete information.

We will assume complete information roughly until the midterm.


The Normal Form

A convenient way to represent a two-player simultaneous move gameof complete information is through the normal form (also known asstrategic form).

Prof.Give Quiz No Quiz

Bob Go to class 0,0 2,6Skip class -5,-1 5,4

By convention, player 1 (Bob) picks the row, and player 2 (professor)picks the column.

Each cell gives the payoffs of player 1 and player 2, in that order.

What do you expect the professor to do? What about Bob?


Dominance

For now, let’s only consider pure strategies.

A player’s strategy is (strictly) dominant if, for any combination ofactions by other players, it gives that player a strictly higher payoffthan all her other strategies. ("The unique best choice no matterwhat others do")

Examples: professor not giving a quiz.

A player’s strategy is (strictly) dominated if there exists anotherstrategy giving that player a strictly higher payoff for all combinationsof actions by other players. ("There’s something else that’s alwaysstrictly better")

Examples: professor giving a quiz.

If a strategy is dominant, what can you say about that player’s otherstrategies?

Is the converse true?


Dominance Solvability (I)

It makes sense to predict that a player will play her dominant strategyif she has one, and will never play a dominated strategy.

But in our example, while we can predict what the professor does,Bob doesn’t have dominant or dominated strategies.

Idea: take it a step further, and assume Bob can also deduce whatthe professor does.

Now we can predict what Bob will do.


Dominance Solvability (II)

Iterated deletion of strictly dominated strategies, or iterated strictdominance (ISD): after deleting dominated strategies, look atwhether other strategies became dominated with respect to theremaining strategies. If so, delete these newly dominated strategies,and repeat the process until no strategy is dominated.

Example: "Going to class" is initially not dominated, but after "Givequiz" is eliminated, it becomes dominated.

A game is dominance solvable if ISD leads to a unique predictedoutcome, i.e. only one strategy for each player survives.

The Bob/Professor game is dominance solvable.


Another Example

Cooperate DefectCooperate -1,-1 -10,0Defect 0,-10 -8,-8

This is an example of the famous Prisoner’s Dilemma."Cooperate" = keeping quiet; "Defect" = rat out the other prisoner

Is this game dominance solvable?

Is your predicted outcome Pareto effi cient?

Defecting gives both players a higher payoff no matter what the otherone does, but (Defect, Defect) is worse for both than (Cooperate,Cooperate).


The Prisoner’s Dilemma

In general, a PD is a 2x2 game where:1 Both players have a dominant strategy.2 The outcome where both players play their dominated pure strategy isstrictly better for both players than the outcome where they play theirdominant strategy.

Lesson: In some games, the outcome that you expect to occuris ineffi cient.Can you think of other situations that can be modeled as a PD?


Exercise

Find the set of strategies that survive ISD.Left Center Right

Top 1,7 1,1 7,0Middle 5,3 6,4 5,1Bottom 3,0 6,5 6,0


Remarks on Dominance Solvability

A solution through ISD is not as convincing as a solution in dominantstrategies: it assumes that players correctly anticipate what otherswill do.

However, ISD allows us to solve some games that don’t have adominant strategy for all players.

Moreover, the type of reasoning carried out in ISD seems feasible andrealistic, especially when the number of steps is small. So it’s still apretty appealing concept.

Fact: the order in which ISD is carried out (which can vary sincethere can be more than one dominated strategy at a time) does notinfluence which strategies survive. (Brainteaser for math jocks: provethis.)


Many Games Are Not Dominance Solvable

It’s easy to come up with a game where ISD doesn’t help at all.ColinPond Timmie’s

Rowena Pond 1,1 0,0Timmie’s 0,0 1,1

(This is a coordination game.)Are we completely stuck?


Nash Equilibrium (Motivation)

If Rowena and Colin haven’t set a meeting place, are meeting eachother for the first time and are new at SFU, then it’d be hard topredict their behaviour.

But if they have communicated, have done this before or if there’s asocial norm, it seems likely that they will coordinate successfully.

Idea: We expect situations where everybody is playing her beststrategy, given others’strategies.

This way, nobody is making a mistake or has an incentive to switch.

Note: Just like for ISD, we are assuming that each player has correctbeliefs about what others are doing. But it’s a stronger assumptionhere, because logic alone doesn’t allow us (or the players) to predictbehaviour.


Nash Equilibrium (Definition)

A player i’s strategy si is a best response to other players’strategiesif, taking as fixed these other players’strategies, si gives player i herhighest possible expected payoff.

In other words, si is a best response if, given what others are doing,no other strategy yields a strictly higher expected payoff for player i .

A Nash equilibrium (NE) is a strategy profile where everyplayer’s strategy is a best response to other players’strategies.Again, this means that nobody has a reason to switch, giving whateveryone else is doing.

An NE in pure strategies or a pure-strategy NE is an NE whereevery player’s strategy is pure.

Examples: (Skip Class, No Quiz) in first example; (Confess, Confess)in PD; (Middle, Center) and (Bottom, Center) in exercise; (Pond,Pond) and (Timmie’s, Timmie’s) in last example.


Finding Pure-Strategy Nash Equilibria

Let’s find all the pure-strategy NEs in the following game:Left Center Right Palin

Top 0,1 0,0 5,2 -10,-10Middle 1,1 4,3 4,0 -10,-10Bottom 2,3 6,2 3,3 -10,-10


Exercise

Find all the pure-strategy NEs in the following game:Left Center Right

Top 3,7 9,6 1,6Middle 0,0 4,0 2,0Bottom 4,0 1,5 0,1


Relating Dominance Solvability and Nash Equilibrium

Does a combination of strategies surviving ISD have to be a NE?

Can a player’s strategy in a NE involve an action eliminated by ISD?

Based on the above, which concept yields sharper predictions? Doesyour answer depend on the game?


Some Games Have No Pure-Strategy NE (I)

Rock Paper ScissorsRock 0,0 -1,1 1,-1Paper 1,-1 0,0 -1,1Scissors -1,1 1,-1 0,0But there might still be a mixed-strategy NE.


Some Games Have No Pure-Strategy NE (II)

We saw that Rock-Paper-Scissors has no pure-strategy NE. Here’sanother example:

KickerLeft Right

Goalie Left 1,-1 -1,1Right -1,1 1,-1

(This game is sometimes called "matching pennies".)What do soccer players actually do?

These are games where it’s important to keep the other(s) guessing.


Optimality of Mixed Strategies

KickerLeft Right


What would the goalie do if the kicker is more likely to play Left?Right?

When will the goalie play both Left and Right with positiveprobability?

In a NE, every action played with positive probability must be abest response.So if a player plays multiple actions with positive probability, hemust be indifferent between them, i.e. they must yield thesame expected utility.


Finding Mixed-Strategy NE

Unfortunately, it is generally hard to find all mixed-strategy NE.

But in 2x2 games (2 players and 2 actions for each player), it’s nottoo bad.

KickerLeft Right


Suppose the goalie plays Left with probability p. To find the NE, weneed to find the value of p that makes the kicker indifferent betweenLeft and Right.

Similarly, we need to find the mix of the kicker’s actions that makesthe goalie indifferent between Left and Right.


(Politically Incorrect) Example: Battle of the Sexes

Find all NE in the following game:

GuyBallet Hockey

Girl Ballet 3,1 0,0Hockey 0,0 1,4

We can also find the expected payoffs of each NE.

Note: for games with a finite number of players and where eachplayer has a finite number of actions, a NE always exists!


Bigger Games

Remember that NEs cannot involve strategies that are eliminated byISD (this is true of all NEs, not just pure-strategy ones).

So finding all mixed-strategy NE is also feasible in games that reducedown to 2x2 through ISD.

In games that don’t reduce that far, sometimes payoffs are nice andsymmetric like in Rock-Paper-Scissors. But even then, it’s more workthan for 2x2 games.

Exercise (to do at home): Find all NE in RPS.


Visualizing NE in 2x2 Games

We can graphically represent players’best responses (BR).

GuyBallet Hockey

Girl Ballet 3,1 0,0Hockey 0,0 1,4

Let p be the probability of "Ballet" for the girl, and q be theprobability of "Ballet" for the guy.

What is the guy’s BR if p < 0.8? If p > 0.8? If p = 0.8?

What is the girl’s BR if q < 0.25? If q > 0.25? If q = 0.25?

The NE correspond to the intersection of the BR curves.


Mixed Strategies and ISD (I)

Remember how we defined ISD: after deleting dominated strategies,look at whether other strategies became dominated with respect tothe remaining strategies. If so, delete these newly dominatedstrategies, and repeat the process until no strategy is dominated.

No mention that the strategies must be pure!

Here are some potential issues:1 We’ve been deleting pure strategies. But there are mixed strategiesthat place positive probability on those pure strategies. So can wereally ignore the strategies that have been deleted?

2 We’ve been deleting a strategy si if there’s another strategy s ′i thatbeats si for any pure strategy by the other player. But we’re onlysupposed to delete si if we can find an s ′i that beats si for any strategy,pure or not, by the other player. Are we "overdeleting"?

3 We’ve been only deleting a strategy if it’s dominated by another purestrategy. But we’re supposed to delete strategies that are dominated byany strategy, pure or not. Are we "underdeleting"?


Mixed Strategies and ISD (II)

1 If pure strategy si is dominated by pure strategy s ′i , then any strategythat uses si with positive probability is dominated by a strategy thatshifts that probability to s ′i . Therefore, when we delete a purestrategy si , we are also automatically deleting any mixedstrategy that places positive probability on that pure strategysi , so we can really safely ignore si .

2 If s ′i gives a higher payoff than si for any pure strategy by the otherplayer, then it also gives a higher expected payoff for any potentiallymixed strategy by the other player: a weighted average is higher if allthe numbers that you’re averaging are higher (and the probabilitydistribution - i.e. the other player’s mixed strategy - stays the same).Therefore, si really is dominated, and we are not overdeleting.

3 Yes, we are underdeleting if we only consider pure strategies:see problem set.


Mixed Strategies and ISD (III)

Note: The observations on the previous slide all carry through withmore than two players, if instead of talking about "the other player’sstrategies," we talk about "the other players’strategy combinations"(i.e. partial strategy profiles that only include the others’strategies).

Note: We’re also "underdeleting" in another way: sometimes, amixed strategy can be dominated even if the pure strategies on whichit places positive probability are not dominated. But this is lessworrisome because it doesn’t make us underdelete pure strategies,which means that it doesn’t "block" later steps of the ISD procedure.


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