lecture 3 - normal form games

Introduction to Game Theory: Normal Form GamesEconomics 302 - Microeconomic Theory II: Strategic Behavior

Instructor: Songzi Du

compiled by Shih En LuSimon Fraser University

January 15, 2015

ECON 302 (SFU) Lecture 3 January 15, 2015 1 / 39

Why Games?

We want to model strategic behavior conscious behavior arisingamong a small number of competitors or players, in a situation whereall are aware of their (possibly conflicting) interests andinterdependence of their decisions.

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

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


What is a Game?

A game has players that each chooses from actions available tohim/her (given his/her information).

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

An outcome is a collection of actions taken by each 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 playerspayoff from an uncertain situation is just the weighted average of herpayoffs from each outcome (where weights = probabilities).


Simultaneous-Move Games of Complete Information

We first study simultaneous-move games of complete information.

Simultaneous-move means that each player picks her action withoutknowing others, and each player moves once.

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

Complete information means that each player knows every playerspayoff 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.


What is a Strategy?

In a game, each player plays a strategy, which in the case ofsimultaneous-move game of complete information is a probabilitydistribution over her actions

A strategy specifies how the player will play his/her game.

Example: Go to class is one of Bobs strategies. Go to class withprobability 0.3 and Skip class with probability 0.7 is another.

A pure strategy is a strategy that puts all weight on an action(probability 1).

A mixed strategy is just any strategy. Mixed is used to emphasizethat the strategy may not be pure.

A collection of each players strategy is called a strategy profile.

Example: (Go to class, 0.4 Give quiz + 0.6 No quiz) is a strategyprofile.


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, or game matrix).

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?


Example: The Prisoners Dilemma (PD)

Not Guilty (Cooperate) Guilty (Defect)Not Guilty (Cooperate) -2, -2 -5, -1

Guilty (Defect) -1, -5 -3, -3ECON 302 (SFU) Lecture 3 January 15, 2015 7 / 39

Example: The Prisoners Dilemma (II)

What do you expect each prisoner to do?

What is the Pareto efficient outcome in the Prisoners Dilemma (PD)?

In a PD, defecting gives both players a higher payoff no matter whatthe other one does, but (Defect, Defect) is worse for both than(Cooperate, Cooperate).

Can you think of other situations that can be modeled as a PD?


Dominance

For now, lets only consider pure strategies.A players 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 best choice no matter whatothers do)Examples: prisoner defecting, professor not giving a quiz.

A players strategy is (strictly) dominated if there exists anotherstrategy giving that player a strictly higher payoff for all combinationsof actions by other players. (Theres something else thats alwaysbetter)Examples: prisoner cooperating, professor giving a quiz.

If a strategy is dominant, the players other strategies must bedominated.On the other hand, there may not be a dominant strategy, eventhough some strategy is dominated.


Dominance

For now, lets only consider pure strategies.A players 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 best choice no matter whatothers do)Examples: prisoner defecting, professor not giving a quiz.

A players strategy is (strictly) dominated if there exists anotherstrategy giving that player a strictly higher payoff for all combinationsof actions by other players. (Theres something else thats alwaysbetter)Examples: prisoner cooperating, professor giving a quiz.

If a strategy is dominant, the players other strategies must bedominated.On the other hand, there may not be a dominant strategy, eventhough some strategy is dominated.ECON 302 (SFU) Lecture 3 January 15, 2015 9 / 39

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.

This allows us to predict the outcome in the PD.

But in our first example, while we can predict what the professordoes, Bob doesnt have dominant or dominated strategies.

Idea: take it a step further, and assume Bob can also deduce whatthe professor does (plays his dominant strategy, which is not givingthe quiz).

Now we can predict what Bob will do (skip class).


Dominance Solvability (II)

Iterated deletion of strict 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. Therefore, we eliminateGoing to class in the second iteration.

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

Both our examples up to now are dominance solvable.


Exercise

Find the set of strategies that survive ISD.

Left Center RightTop 1, 7 1, 1 7, 0

Middle 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 not do (i.e., how the others will eliminate their actions).

However, ISD allows us to solve some games that dont have adominant strategy for all players.

Moreover, the type of reasoning carried out in ISD seems feasible andrealistic, especially when the number of iterations is small. So its stilla pretty 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.)


Travelers Dilemma

An airline has lost two suitcases belonging to Alice and Bob. Identicalantiques in both suitcases.

Each customer writes a claim value, simultaneously:

2 vA, vB n,where n is the maximum possible value.

If both claim the same value, each gets that amount.

If Alice claims a smaller value (vA < vB), Alice gets vA + 2, Bob getsvA 2.

If Bob claims a smaller value (vB < vA), Alice gets vB 2, Bob getsvB + 2.


Travelers Dilemma (II)

For simplicity, suppose n = 4. Write down the normal form.

The strategy of claiming 4 is not strictly dominated by any other purestrategy.

Nevertheless, the strategy 4 is strictly dominated by the mixedstrategy consisting of claiming 3 with probability 1/2 and claiming2 with probability 1/2.

Therefore, we can eliminate the (strictly dominated) strategy ofclaiming 4.

After the strategy of claiming 4 is eliminated, strategy of claiming 3 isstrictly dominated by the strategy of claiming 2.

Therefore, a single strategy survives ISD: claiming 2 (for each player).

Exercise: write down the normal form for the travelers dilemmaswith n = 5 and n = 6. Solve by ISD. Extra challenge: n = 100.


Mixed Strategies and ISD

Some pure strategies may only be strictly dominated by mixedstrategies.

A players strategy is strictly dominated if there exists anotherstrategy, pure or mixed, giving that player a strictly higher payoff forall combinations of strategies by other players. (Theres somethingelse thats always better)

A pure strategy that is strictly dominated by a mixed strategy need tobe (iteratively) eliminated in ISD.


Many Games Are Not Dominance Solvable

Its easy to come up with a game where ISD doesnt help at all.

ColinPond Timmies

Rowena Pond 1, 1 0, 0Timmies 0, 0 1, 1

(This is a coordination game.)

Are we completely stuck?


Nash Equilibrium (I)

If Rowena and Colin havent set a meeting place, are meeting eachother for the first time and are new at SFU, then itd be hard topredict their behavior.

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

Idea: We expect situations where everybody is playing her beststrategy, given (that he/she correctly anticipates) 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 its a stronger assumptionhere, because logic alone doesnt allow us (or the players) to predictbehaviour.


Nash Equilibrium (II)

A player i s (pure or mixed) strategy i is a best response to otherplayers strategies if, taking as fixed these other players strategies, igives player i her highest possible payoff.

A Nash equilibrium (NE) is a strategy profile (1, . . . , n) whereevery players strategy is a best response to other players strategies.

Again, this means that nobody has a reason to switch (no incentiveto deviate), giving what everyone else is doing.

An NE in pure strategies or a pure-strategy NE is an NE whereevery players strategy is pure.

Examples of pure-strategy NE: (Pond, Pond) and (Timmies,Timmies) in the last example.


Finding Pure-Strategy Nash Equilibria

Lets find all the pure-strategy NEs in the following game:

Left Center Right DumbTop 0, 1 0, 0 5, 2 -10, -10

Middle 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 RightTop 3, 7 9, 6 1, 6

Middle 0, 0 4, 0 2, 0Bottom 4, 0 1, 5 0, 1


Some Games Have No Pure-Strategy Nash Equilibria

Rock Paper ScissorsRock 0, 0 -1, 1 1, -1Paper 1, -1 0, 0 -1, 1

Scissors -1, 1 1, -1 0, 0

But there might still be a mixed-strategy NE.


Some Games Have No Pure-Strategy NE

We saw that Rock-Paper-Scissors has no pure-strategy NE. Heresanother example:

KickerLeft Right

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

What do soccer players actually do?

These are games where its 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?

If a player mixes between multiple pure strategies in a NE, then all ofthese strategies (the ones played with positive probability) must bebest responses.


Finding Mixed-Strategy NE

General procedure to find a Mixed-Strategy NE:1 For each player, conjecture a set of strategies that he/she mixes over.2 Calculate the mixing probabilities so that no one has incentive to

deviate (Nash Equilibrium).

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

But in 2x2x2 games (2 players and 2 actions for each player), its nottoo bad.


Kicker-Goalie Revisited

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 kickers actions (playing Leftwith probability q) that makes the goalie indifferent between Left andRight.


(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 compare the expected payoffs of NE.


Another example: mixing over some (but not all) strategies

Find all NE in the following game:

D E F

A 4, 0 0, 4 3, 0

B 0, 4 4, 0 3, 0

C 0, 3 0, 3 3, 3


Nashs Theorem

Theorem (Nash, 1950)

For a game with a finite number of players and where each player has afinite number of actions, a Nash Equilibrium always exists.

John F. Nash (born 1928)

PhD in mathemaitcs, Princeton (1950)

won the Nobel Prize in Economics, 1994

movie A Beautiful Mind


Relating Dominance Solvability and Nash Equilibrium

The support of a NE always lies among ISD strategies:

For NE (1, 2, . . . , n), if 1 places positive probability on a strategys1, s1 must be player 1s best response given 2, . . . , n.

Therefore, s1 cannot be strictly dominated. (strictly dominated nota best response for anything)

Therefore, strategies in the support of i are never deleted during theISD iterations.

NE offers sharper prediction than ISD.


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 2x2x2 through ISD.

In games that dont reduce that far, sometimes payoffs are nice andsymmetric like in Rock-Paper-Scissors. But even then, its more workthan for 2x2x2 games.


Symmetry

A symmetric game is a game where the payoffs for playing aparticular strategy depend only on the other strategies employed, noton who is playing them.

If one can change the identities of the players without changing thepayoff to the strategies, then a game is symmetric.Examples: prisoners dilemma, travelers dilemma, rock-paper-scissor,etc.

X YX a, a c, dY d, c b, b

A symmetric Nash equilibrium is a Nash equilibrium where everyplayer plays the same strategy.

A symmetric game always has a symmetric Nash equilibrium, though itmay also process an asymmetric Nash equilibrium.For a symmetric Nash equilibrium: only needs to check that a singleplayer has no incentive to deviate.


Symmetric mixed-strategy NE

Rock Paper ScissorsRock 0, 0 -1, 1 1, -1Paper 1, -1 0, 0 -1, 1

Scissors -1, 1 1, -1 0, 0

Suppose that each player plays rock with probability r , paper withprobability p, and scissor with probability s. Find the mix-strategyNash equilibrium.


Understanding the Bystander Effect

Motivation:

Thirty-eight people witnessed the brutal murder of Catherine (Kitty)Genovese over a period of half an hour in New York City in March1964.

During this period, none of them significantly responded to her screamsfor help; none even called the police.

A crime is observed by n people.

Each person would like the police to be informed (value v), but prefersthat someone else make the phone call (cost c). Assume v > c > 0.

Action of a person: call the police, or dont call the police.

Note that we may reinterpret (relabel) calling the police as helpingthe victim.


Understanding the Bystander Effect (II)

Pure strategy: 1 person calls the police, the rest do not call.

Symmetric mixed strategy: each person calls police with probability p.

v(1 (1 p)n1) = v c

p = 1 (c/v) 1n1 .Probability p that an individual calls the police decreases with thegroup size n.


Understanding the Bystander Effect (III)

What is the probability the crime is reported in equilibrium?

The probability that no one calls the police is

(1 p)n = (c/v) nn1 = (c/v)1+ 1n1 ,since p = 1 (c/v) 1n1 .Alternatively,

(1 p)n = (1 p)n1 (1 p) = cv

(1 p)because v(1 (1 p)n1) = v c from the equilibrium condition.In any case, the probability that the crime is unreported increaseswith the number n of bystanders.

Bystander effect: more bystanders lead to less probability ofreporting the crime/helping the victim.


Social Psychology vs. Game Theory (Osborne, 2003)

Social psychology:

1 diffusion of responsibility: the larger the group, the lower thepsychological cost of not helping.

2 audience inhibition: the larger the group, the greater theembarrassment suffered by a helper in case the event turns out to beone in which help is inappropriate.

3 social influence: a person infers the appropriateness of helping fromothers behavior, so that in a large group everyone elses lack ofintervention leads any given person to think intervention is less likely tobe appropriate.

These explanations amount to increasing the cost c and decreasingthe benefit v (of intevening/reporting the crime) with n.


Social Psychology vs. Game Theory (II)

We have instead assumed that the cost c and the benefit v areconstant, independent of group size n. We conclude that thebystander effect is something more fundamental, an inevitableconsequence of the strategic interaction of the bystanders.

Our Nash equilibrium analysis rests on the premise that whether aperson intervenes depends on how likely he/she thinks others wouldintervene. This perfectly reasonable premise leads to the disturbingconclusion of the bystander effect.

Advantage: game-theoretic analysis is universal, the same analysis isapplicable to the bystander effect as well as to market competition,voting, bargaining, etc.


Conceptual Review Questions

1 Is pure strategy a mixed strategy? And the converse?2 Whats the difference between a strategy profile and a Nash

equilibrium?3 In a mixed-strategy Nash equilibrium, why solve for indifference? Why

not do it for a pure-strategy Nash equilibrium?4 Does it ever make sense to compare different players payoffs?5 When youre doing ISD, do comparisons between player 2s payoffs

help determine whether you should delete a row?6 Why can a NE, whether or not it is in pure strategies, only involve

strategies that survive ISD?7 If a game is dominance solvable, what can you say about its NE?8 If you need to find the NE of a game thats bigger than 2x2x2, what

could you do first to reduce your workload?9 Is game theory only about complete-information and

simultaneous-move games?


lecture 3 - normal form games

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