EconS 305 - Game Theory - Part 2Again, we apply backward induction. When the husband selects...

EconS 305 - Game Theory - Part 2

Eric Dunaway

Washington State University

eric.dunaway@wsu.edu

November 12, 2015

Eric Dunaway (WSU) EconS 305 - Lecture 29 November 12, 2015 1 / 47

Introduction

Yesterday, we covered simultaneous move games and found out howto determine Nash Equilibria.

Today, we will look at sequential move games, and see how ourequilibria change when the order of movement matters.

Sequential Move Games

The big di¤erence between sequential and simultaneous move gamesis that now, the players do not move at the same time.

If you think about it, all of the rules we have for simultaneous gamesstill apply, we are just adding in some new information to use.

Let�s look at a familiar example: The Prisoner�s Dilemma!

Silence

Betray

Silence Betray

Player 1

Player 2

1 1 5 0

0 5 3 3

The setup is exactly the same, we just now have player 1 (who Ipicked arbitrarily) move �rst. Player 2 gets to oberve player 1�s move,then make his move.

We can draw this like we did before with the normal form. This newform is called the extensive form.

Player 1

Player 2Player 2

Silence Betray

Silence Betray Silence Betray

A few things to notice:

Player 1 still has only one move to make and one action to choose.Player 2 now has two di¤erent actions to take, one for each of thepossible moves that player 1 makes. We call each of player 2�s nodes aseperate information set.With two actions, player 2�s strategies will now consist of pairs ofactions.

To solve this game, we use a technique known as backward induction.

We start at the bottom of the extensive form game tree and �gure outthe best response of the moving player. Then we move up the treeuntil we get back to the origin.The equilibrium will be the strategies that follow the path down thetree.

Player 1

Player 2Player 2

Silence Betray

Player 2

Silence Betray

Player 2

Silence Betray

Player 2

Silence Betray

Player 2

Silence Betray

Player 1

Player 2Player 2

Silence Betray

Upon observing player 1 play "Silence," player 2�s best response is toplay "Betray."

Likewise, upon observing player 1 play "Betray," player 2�s bestresponse is to play "Betray."

For simplicity, we can just substitute these outcomes up to player 1�sactions.

This is known as the reduced form game.

After that, we just perform the same process for player 1.

Player 1

Player 2Player 2

Silence Betray

Player 1

Silence Betray

Player 1

Silence Betray

Since player 1 knows that player 2 is going to pick "Betray"regardless of his actions, he knows that it is better for him to play"Betray" �rst, as that gives him the highest payo¤.

Now, we insert this result into our original tree to get our NashEquilibrium.

Player 1

Player 2Player 2

Silence Betray

As we can see, the sequential game has the same result as thesimultaneous game.

This is of no coincidence. When a game only has one Nash Equilibrium,the simultaneous and sequential games will have the same result.

Furthermore, we could redo this game having Player 2 move �rst, andwe would also end up at the same result.

This, however, is not always the case. This is a result from both player1 and player 2 having the same payo¤ structure.Let�s look at an example where the order matters.

Battle of the Sexes

Fight Opera

Husband

1 3 0 0

0 0 3 1

Battle of the Sexes

The setup of the game is the same as in the simultaneous version,only this time, the husband gets to select where he is going �rst.

The wife can then observe the husband�s choice and then make herown choice.

Battle of the Sexes

Husband

WifeWife

Fight Opera

Fight Opera Fight Opera

Battle of the Sexes

Again, we apply backward induction.

When the husband selects "Fight," the wife�s best response is also"Fight."

When the husband selects "Opera," the wife�s best response is also toselect "Opera."

Battle of the Sexes

Husband

WifeWife

Fight Opera

Battle of the Sexes

Moving up the tree, since the husband knows the wife�s best responsefunctions, he can select "Fight" and end up with a payo¤ of 1, or hecan select "Opera" and end up with a payo¤ of 3.

Naturally, he will pick Opera.

Battle of the Sexes

Husband

WifeWife

Fight Opera

Battle of the Sexes

This gives us an equilibrium prediction where both the husband andthe wife go to the opera.

This is actually quite nice. The simultaneous game had two possibleNash Equilibria, but this game only has one!Why? The wife might prefer to go to the �ght, but she is rational, andthe husband knows this. He knows that she prefers to go to the sameevent as he does and will thus select whichever even he chooses to goto �rst. Taking advantage of this, he selects the opera.The wife could try to hold out and tell her husband that she is going togo to the �ght, but the husband knows better that she is rational.Were she to do this, it would be known as an incredible threat.

Battle of the Sexes

Will this same result hold if we let the wife move �rst?

Now, the husband wanting to go to the opera would be an incrediblethreat.Let�s see what happens.

Battle of the Sexes

HusbandHusband

Fight Opera

Battle of the Sexes

HusbandHusband

Fight Opera

Battle of the Sexes

HusbandHusband

Fight Opera

Battle of the Sexes

Now the result is the opposite!

Since the wife knows that the husband is rational (as well as his bestresponse functions), she can choose to go to the �ght �rst, and thehusband will respond by also going to the �ght.This is the other Nash Equilibrium as predicted by the simultaneousmove game.

Battle of the Sexes

In sequential move games like the Battle of the Sexes, the order ofmovement matters.

It will determine which of the simultaneous move Nash Equilibria isselected.In these cases, we would say that this equilibrium is sequentiallyrational, and that the equilibrium is called a subgame perfect NashEquilibrium.

That�s essentially the process for solving sequential move games.

On a quiz, I might ask you to solve games that have three levels, ormaybe three possible actions at a node.The steps are the same! Just apply backward induction and work yourway back up the tree.Let�s try one!

Example

Player 1

Player 1 Player 1

Player 2Player 2

Example

In this game, player 1 moves, then player 2 moves, then, dependingon player 2�s action, player 1 might get to move one more time!

Nothing has changed though; we still start at the bottom of our gametree and work our way back up through backward induction.

In this case, we start with player 1�s actions at the bottom of thegame tree.

Example

Player 1

Player 1 Player 1

Player 2Player 2

Example

Player 1

Player 1 Player 1

Player 2Player 2

Example

Player 1

Player 1 Player 1

Player 2Player 2

Example

Player 1

Player 1 Player 1

Player 2Player 2

Example

As we found, in equilibrium, �rst player 1 moves Left (L)

Then, player 2 also moves Left (L)Lastly, player 1 gets to move again and chooses Right (R)

Player 1 receives a payo¤ of 5 and player 2 receives a payo¤ of 2.

Summary

Sequential move games add information on top of their simultanousmove counterparts.

They can resolve issues where multiple Nash Equilibrium predictionswere found before.

We will be using several Game Theory concepts in our next units.

Preview for Monday

Applications of Game Theory.

BargainingThe Boardwalk GameSome Industrial Organization

Some Practice (1 of 1)

1. On the next slide, I have drawn the normal form game where two�rms are planning to sell either 10 or 20 units of their goods andreceive pro�t levels outlined in the normal form game.

a. What is the Nash Equilibrium(s) if both �rms make their decisionsimultanously?

b. Draw the extensive form game tree if Firm 1 can decide �rst. What isthe outcome? Why?

c. Draw the extensive form game tree if Firm 2 can decide �rst. What isthe outcome? Why?

Some Practice (1 of 1)

Firm 1

Firm 2

30 30 50 35

60 40 20 20

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