Markets The Economics of Platform Competition and Online Markets

Econ 191:The

Economics ofPlatform

Competitionand OnlineMarkets

Preparatorylecture forProf. JohnMorgan

Issi Romem

Introduction

Nashequilibriumwith purestrategies

Nashequilibriumwith mixedstrategies

Experimentaldesign: theimportance ofrandomization

Econ 191:The Economics of Platform Competition

and Online MarketsPreparatory lecture for Prof. John Morgan

Issi Romem

February 21, 2012

Econ 191:The

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Introduction

Today’s lecture:

Nash equilibrium with pure strategies

Nash equilibrium with mixed strategies

And also:

The importance of randomization in experimental design.

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Today’s lecture:

And also:

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Today’s lecture:

And also:

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Today’s lecture:

And also:

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What is a “game”?

Game theory is a mathematical tool that can help understandsituations in which decision-makers interact.What constitutes a game?

Decision-makers, called “players”.

For each player, a set of possible “strategies”.

For each possible combination of players’ strategies, an“outcome”, consisting of a “payoff” for each player.

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Game theory is a mathematical tool that can help understandsituations in which decision-makers interact.

What constitutes a game?

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Example:

Jacktv internet

Jilltv 2,2 5,5

internet 6,6 0,0

Two players in this game, Jack and Jill.

Jack has two strategies, tv and internet, and so does Jill.

The outcomes for each combination of players’ strategiesis given.

What will happen?

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Example:

Jacktv internet

Jilltv 2,2 5,5

internet 6,6 0,0

What will happen?

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Example:

Jacktv internet

Jilltv 2,2 5,5

internet 6,6 0,0

What will happen?

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Example:

Jacktv internet

Jilltv 2,2 5,5

internet 6,6 0,0

What will happen?

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Example:

Jacktv internet

Jilltv 2,2 5,5

internet 6,6 0,0

What will happen?

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Example:

Jacktv internet

Jilltv 2,2 5,5

internet 6,6 0,0

What will happen?

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In order to say what will happen, we need a “solution concept”.

“Nash equlibrium” is a solution concept, and can be defined asfollows:

Nash equilibrium

An outcome is a Nash equilibrium if, given the strategies of allother players, no player has anything to gain by changing onlyhis own strategy unilaterally.

What outcome/s are a Nash equilibrium in the previous game?

Econ 191:The

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Nash equilibrium

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Nash equilibrium

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Nash equilibrium

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Nash equilibrium

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Examples:

Jack and Jill:

Player 2: Jacktv internet

Player 1: Jilltv 2,2 5,5

internet 6,6 0,0

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Examples:

Coordination game:

AT&T Verizon

AT&T 2,2 0,0Verizon 0,0 1,1

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Examples:

Still a coordination game?

T 2,2 0,0B 0,0 -1,-1

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Examples:

The prisoner’s dilemma:

be quiet tell

be quiet 2,2 0,3tell 3,0 1,1

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Examples:

Working together:

work hard good off

work hard 2,2 0,3goof off 3,0 1,1

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Examples:

Hawk/Dove:

H 0,0 3,1D 1,3 2,2

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Examples:

Matching pennies:

Each player has a penny.

Both players simultaneously show each other one face oftheir penny.

Player 1 wins if both pennies show the same face.

Player 2 wins if the pennies show different faces.

Head Tail

Head 1,-1 -1,1Tail -1,1 1,-1

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Examples:

Matching pennies:

Head Tail

Head 1,-1 -1,1Tail -1,1 1,-1

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Examples:

Matching pennies:

Head Tail

Head 1,-1 -1,1Tail -1,1 1,-1

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Examples:

Matching pennies:

Head Tail

Head 1,-1 -1,1Tail -1,1 1,-1

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Examples:

Matching pennies:

Head Tail

Head 1,-1 -1,1Tail -1,1 1,-1

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Examples:

Matching pennies:

Head Tail

Head 1,-1 -1,1Tail -1,1 1,-1

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Does “matching pennies” have a Nash equilibrium?

“Matching pennies” does not have a Nash equilibrium withpure strategies.

What if we define players’ possible strategies differently...

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Mixed strategies

Instead of having just two strategies, Head or Tail, each playercan choose a probability of Head or Tail as his strategy - amixed strategy.

Player 1’s (2’s) strategy is now:Head with probability 0 ≤ p ≤ 1 (0 ≤ q ≤ 1).(Of course, this implies Tail with probability 1− p (1− q).)

This means that each player has a set of strategies that isinfinite!

First, let’s practice finding mixed strategy Nash equilibria.Then, we’ll think about what they actually represent.

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Mixed strategies

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Mixed strategies

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Mixed strategies

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Mixed strategies

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

If player 1 picks the strategy p = 1 (i.e. play Head forsure) then his payoff is: q · 1 + (1− q) · −1 = 2q − 1.

If player 1 picks the strategy p = 0 (i.e. play Tail for sure)then his payoff is: q · −1 + (1− q) · 1 = 1− 2q.

Generally, if player 1 picks the strategy p then his payoff is:p · (2q − 1) + (1− p) · (1− 2q).

What does player 1 do?

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

⇒ Expected payoff 2q − 1

⇒ Expected payoff 1− 2q

2q − 1 > 1− 2q

4q > 2

q > 0.5

If q > 0.5, player 1 chooses p = 1.

If q < 0.5, player 1 chooses p = 0.

If q = 0.5, player 1 is... indifferent, andmay choose any p!

What does player 2 do? A similarcalculation...

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

2q − 1 > 1− 2q

4q > 2

q > 0.5

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

2q − 1 > 1− 2q

4q > 2

q > 0.5

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

2q − 1 > 1− 2q

4q > 2

q > 0.5

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

2q − 1 > 1− 2q

4q > 2

q > 0.5

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

2q − 1 > 1− 2q

4q > 2

q > 0.5

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

Player 1: q > 0.5⇒ p = 1; q < 0.5⇒ p = 0;q = 0.5⇒ indifferent, so any p is optimal.

Player 2: p < 0.5⇒ q = 1; p > 0.5⇒ q = 0;p = 0.5⇒ indifferent, so any q is optimal.

⇒ Mixed strategy Nash equilibrium: (p, q) = (0.5, 0.5).

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

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Matching pennies

q 1-qHead Tail

p Head 1,-1 -1,11-p Tail -1,1 1,-1

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Mixture implies indifference between pure strategies

A mixture of two (or more) pure strategies can only be optimalif, given the other players’ behavior, the pure strategies providethe same payoff.

Why?Because if one of the pure strategies provides a bigger payoff, itis optimal to place more weight on it.

We can use this insight to solve for mixed strategies.

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Coordination game revisited

q 1-qAT&T Verizon

p AT&T 2,2 0,01-p Verizon 0,0 1,1

We saw earlier that this game has two pure strategy Nashequilibria:(p, q) = (1, 1) and (p, q) = (0, 0).

Does this game also have any mixed strategy Nashequilibria?

Try it...

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q 1-qAT&T Verizon

p AT&T 2,2 0,01-p Verizon 0,0 1,1

Try it...

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q 1-qAT&T Verizon

p AT&T 2,2 0,01-p Verizon 0,0 1,1

Try it...

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q 1-qAT&T Verizon

p AT&T 2,2 0,01-p Verizon 0,0 1,1

Player 1:

AT&T exp. payoff

= Verzion exp. payoff

2q = 1− q

3q = 1

q = 1/3

Player 2:

AT&T exp. payoff

2p = 1− p

3p = 1

p = 1/3

⇒ A third Nash equilibrium: (p, q) = (1/3, 1/3).

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q 1-qAT&T Verizon

p AT&T 2,2 0,01-p Verizon 0,0 1,1

Player 1:

AT&T exp. payoff

2q = 1− q

3q = 1

q = 1/3

Player 2:

AT&T exp. payoff

2p = 1− p

3p = 1

p = 1/3

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q 1-qAT&T Verizon

p AT&T 2,2 0,01-p Verizon 0,0 1,1

Player 1:

AT&T exp. payoff

2q = 1− q

3q = 1

q = 1/3

Player 2:

AT&T exp. payoff

2p = 1− p

3p = 1

p = 1/3

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q 1-qAT&T Verizon

p AT&T 2,2 0,01-p Verizon 0,0 1,1

Player 1:

AT&T exp. payoff

2q = 1− q

3q = 1

q = 1/3

Player 2:

AT&T exp. payoff

2p = 1− p

3p = 1

p = 1/3

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q 1-qAT&T Verizon

p AT&T 2,2 0,01-p Verizon 0,0 1,1

Player 1:

AT&T exp. payoff

2q = 1− q

3q = 1

q = 1/3

Player 2:

AT&T exp. payoff

2p = 1− p

3p = 1

p = 1/3⇒ A third Nash equilibrium: (p, q) = (1/3, 1/3).

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What do mixed strategies represent?

In matching pennies, it makes sense to think of mixedstrategies literally - players might decide on their move bythrowing dice behind their back.

In the coordination game, this seems artificial.

Another way of thinking about mixed strategy Nash equilibriais as a concise description of the pure strategy Nash equilibriain many slightly different versions of the same game(“Harsanyi’s purification”).

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In a 1973 article, (Berkeley Nobel laureate) John Harsanyisuggested (roughly) the following:

Suppose the payoffs of a game are “disturbed”, i.e. theyhave small random fluctuations around the payoffs given inthe game.

It can be shown that for any “disturbed” version of thegame there exists a pure Nash equilibrium.

The mixed strategy Nash equilibrium can be thought of asthe average of many “disturbed” versions of the game.

The equilibrium mixture of strategies reflects thedistribution of the disturbances.

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In the AT&T vs. Verizon coordination game, think of mixedstrategy Nash eq. as follows:

A specific person can only belong to one network - a purestrategy.

Player 2 describes the average of all other people (a shareq belong to AT&T, the rest to Verizon).

For player 1 the game is “disturbed” in such a way that heeither goes AT&T or Verizon.

If player 1 also describes the average of many people, thenthe mixed strategy Nash eq. is just a concise way ofdescribing all the versions of the game at once.

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The conceptual difficulties with mixed strategy Nash eq. aretwofold:

In most situations, a player randomizing his behaviorseems unrealistic.

In a mixed strategy Nash eq. each player is indifferentbetween his mixed strategies, and chooses exactly theprobability that makes the other players indifferent. Whatprevents players from deviating from their strategy?

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The previous line of thought following Harsanyi solves both ofthese issues:

Players are not actually randomizing. They each face a“disturbed” game.

In the “disturbed” each player chooses a pure strategythat is strictly optimal, and so does not deviate.

The mixed strategy Nash eq. can be thought of as aconcise description of many such disturbed games.

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Experimental design: the importance ofrandomization

In economic experiments, you will often hear that people areassigned to “treatment” and “control” groups randomly.

What is the importance of randomization?

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Let’s think through an example:

Suppose we want to measure the effect of an extra hour ofsection in econometrics.

We set up one section with 3 hrs/wk and another sectionwith 2 hrs/wk, and put this information in the coursecatalog.

At the end of the semester, we compare the averagegrades for students in both sections.

Is the difference between the sections the effect of anextra hour of section time?

No. If better (or worse) students disproportionately select intothe longer section, this may drive (confound!) the difference inaverage grades.

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Would randomly assigning students to the two sections help?

One section might still get better students than the other,but the difference will be independent of the extra hour ofsection time, and will disappear as the sections grow larger.

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Another example:

Suppose we work at Facebook and want to measure theeffect of a new feature on time spent on the site.

We place a little button for installing the feature in thecorner of the page, and then compare the average timespent on the site for users who did and did not use thenew feature.

Would you interpret the difference between the groups asbeing caused by the new feature? Why?

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Another example:

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Another example:

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Another example:

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In general:

If units are not randomly assigned to treatment andcontrol groups, non-random selection into groups preventscausal interpretation.

Non-random selection likely reflects unobservedcharacteristics that affect outcomes, and may drive groupdifferences in addition to the treatment.

Randomization - randomized assignment to groups - solvesthis problem.

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In general:

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In general:

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In general:

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Thank you for your attention.

Good night!

Econ 191:The

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Thank you for your attention.

Good night!

Markets The Economics of Platform Competition and Online Markets

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