games people play. 2. sequential move games. in this section we shall learn how to figure out what...

Games People Play.Games People Play.

2. Sequential move games.2. Sequential move games.

In this section we shall learn In this section we shall learn How to figure out what to expect in games with sequential moves.

Techniques for reducing complex games into easily understood ones.

Why it is that the order of moved can matter for the outcome of the game.

How it can be that some games need to be thought of as you playing against yourself!


Sequential Move GamesSequential Move Games

Topics to be coveredTopics to be coveredGame TreesBackwards Induction Tree Pruning Adding PlayersOrder AdvantagesAdding Moves


Game TreesGame Trees

Sometimes called the extensive form of the game.

A game tree allows an intuitive visual representation of the possible choices and payoffs available to the players. Player #1 (Moves first)

Player #2 (Moves second)

Player #2 (Moves second)

Choice a

Choice b

Choice c

Choice d

Choice e

Choicef

Payoffs#1, #2

1 6

6 1

3 3

10 -5

NodesBranches


How to Construct a Game TreeHow to Construct a Game TreeAn Example Happy-Families vs. IBMAn Example Happy-Families vs. IBM

Suppose IBM already produces servers, but Happy-Families currently do not.

If Happy-Families decide to move into this market it will take 2 years of R&D etc.

IBM can attempt to discourage Happy-Families’ entry by launching an advertising campaign today.

IBM have the first move to choose ads or not, hence the split of the tree at the IBM node.

Happy-Families have the second move to enter or not, hence the split at the Happy-Families nodes.

At the terminal nodes we record the payoffs.

The numbers used are calculated by business intelligence.

IBM

Ads

No Ads

H-F

H-F

In

Out

In

Out

PayoffsIBM, H-F

1,1

3,3

2,4

4,2


Game TreesGame TreesA game tree with an information setA game tree with an information set

An information set indicates that the player only knows that she is in that set and no more.

IBM

Ads

No Ads

Happy-Families

Happy-Families

In

Out

In

Out

PayoffsIBM , Happy-Families

1,1

3,3

2,4

4,2


Backwards Induction - Backwards Induction - Thinking Thinking through sequential games by looking ahead and through sequential games by looking ahead and

reasoning back.reasoning back.Suppose, for the sake of argument you have the last move in a game.

You look at all the possible payoffs and make a contingent plan. If I end up here I will do thus, and receive the associated payoff.

In the preceding play your opponent can figure out what you will do in the last stage. She thinks as follows.

For each of her choices she knows that you will make the subsequent choice that maximizes your payoffs.

Given that she knows you will behave in this fashion she then deduces which of your choices you will make after each of her moves.

She can thus evaluate her choices knowing what you will do in reply to each possibility and the payoff she will receive.

She thus places you in the position where your optimal choice will be most advantageous to her.


Yogi on Backwards InductionYogi on Backwards Induction

"If you don't know where you are going, you "If you don't know where you are going, you will wind up somewhere else."will wind up somewhere else." – Yogi – Yogi Berra.Berra.


Sequential move games.Sequential move games.

The gubernatorial race game.The gubernatorial race game.Two politicians, Arnold and Gary are potential gubernatorial candidates. Arnold has declared his candidacy. Gary has to decide whether or not to enter the race after which Arnold decides whether or not to pay for ads. The potential payoffs, Gary’s first, are

• In-Ad {1,1}.• Out-Ad {3,3}.• In-No ad {4,2}.• Out-No ad {2,4}.

Decide whether to enter if you are Gary.


Sequential move gamesSequential move gamesEquilibrium in the gubernatorial race game.Equilibrium in the gubernatorial race game.

If Gary chooses in we see that Arnold will choose no ads and the payoffs will be {4,2}.If Gary chooses out we see that Arnold chooses no ads and the payoffs are {2,4}.So Gary chooses in and Arnold chooses no ads.This is an example of backwards induction.

Arnie

Gary

Ads

No adsIn

Out

1, 1

4, 2

3, 3

2, 4

Gary Arnie,

Ads

No ads

Arnie


Sequential move games.Sequential move games.

Pure strategies.Pure strategies.Moves that you will play with probability one.

Example; If Arnold chooses no ads Gary will choose in.

In this section we shall only consider pure strategies and pure strategy equilibria.


Sequential move games.Sequential move games.Games against yourself!Games against yourself!

In playing games you must think not about what you would like to plan on doing in the future, but also what you will do when the future actually arrives.

This can be thought of as a sequential game between your current and future self in which your current self moves first.

Any game against any number of opponents may have this same feature. You have to look ahead and see how you will play then next time it’s “your move”.


Sequential move games.Sequential move games.Career choice as aCareer choice as a game against yourself!game against yourself!

Current Self

FutureSelf

MedicalSchool

BusinessSchool

community medicine

LawSchool

family practice

neurosurgeon

public prosecutor

politics

corporate law

finance

management

marketing

$50.000/ yr

$150,000/ yr

$500,000/ yr

$50,000/ yr

$35,000-$200,00/ yr

$600,000/ yr

$100,000/ yr

$250,00/ yr

$0-$5m/yr


Sequential move games.Sequential move games.Career choice as aCareer choice as a game against yourself!game against yourself!

As an idealistic 19 year old you might wish to eventually go into community medicine.

But you know that as a 26 year old burdened with debt from graduate school you will choose the highest paying option.

Your choices are really between neurosurgery, corporate law, and finance.


Sequential move games.Sequential move games.A game against yourself - A real world A game against yourself - A real world

exampleexampleAn Happy-Families corporation executive spots an opportunity to change the materials used to manufacture a particular input. By doing so the cost of producing the input will fall significantly. The manufacturer of the input is reluctant to adopt the new material despite the obvious reduction in costs. The reason is that if it adopts the material it will be tied to Happy-Families as its only potential customer. Happy-Families knows this and will subsequently adopt a much more aggressive bargaining posture. Happy-Families’ problem today is to commit itself to being generous with the parts supplier in the future despite the fact that when the future comes it will wish to do otherwise.

Happy-Families today is effectively playing a game against Happy-Families tomorrow!!


Sequential move gamesSequential move gamesThe importance of move sequenceThe importance of move sequence

The gubernatorial race game again.The gubernatorial race game again.Two politicians, Arnold and Gary are again potential gubernatorial candidates. Arnold has declared for the election, but has not decided on whether to launch any campaign ads. This time Arnold get the first move and must decides whether or not to pay for campaign ads. Following this Gary decides whether or not to enter the race. The potential payoffs, Arnold’s first, are as before

• Ad-in {1,1}.• Ad-out {3,3}.• No ad-in {2,4}.• No ad-out {4,2}.

Does the move sequence change the outcome, and if so why?


Sequential move gamesSequential move gamesEquilibrium in the new gubernatorial race game.Equilibrium in the new gubernatorial race game.

If Arnold chooses ads we see that Gary will choose out and the payoffs will be {3,3}.If Arnold chooses no ads Gary chooses in and the payoffs are {2,4}.So Arnold chooses ads and Gary does not enter.

Gary

Arnie

Ads

No ads

In

Out

1, 1

3, 3

2, 4

4, 2

Arnie, Gary

GaryIn

Out


Sequential move gamesSequential move games

The chocolate cake game.The chocolate cake game.Each pair of participants has to divide a chocolate cake.

The first participant divides the cake.

The second chooses which piece to take.

What do you do?


Sequential move gamesSequential move games

The chocolate cake game - equilibrium.The chocolate cake game - equilibrium.Apply backwards induction.

The person choosing the piece will choose the largest one.

The person cutting then has an incentive to maximize the size of the smallest piece.

So, cut the cake in half.


Sequential move gamesSequential move gamesMore than 2 players.More than 2 players.

The logic of solving sequential move games with more than two players is the same as with two players. Apply backwards induction…..think ahead and reason back.One technique that allows us to keep track of this process is often referred to as pruning the tree.Consider three computer firms (surprise!!) Dell, Happy-Families, and Gateway. Each can manufacture either in or out of the US

Each manufacturer can benefit from employing cheap foreign labor and can benefit from learning from the others experience of dealing with foreign regulations, unions, customs and the like. However there is not enough skilled foreign labor for all three firms. The following game might arise.


Sequential move gamesSequential move gamesMore than Two PlayersMore than Two Players

D ell

H P

H P

G atew ay

G atew ay

G atew ay

G atew ay

U S

U S

U S

U S

U S

U S

U S

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

P a y o ffs1 ,5 ,5

5 ,5 ,2

5 ,2 ,5

3 ,4 ,4

2 ,5 ,5

4 ,3 ,4

4 ,4 ,3

4 ,4 ,4

D ell

H P

H P

G atew ay

G atew ay

G atew ay

G atew ay

U S

U S

U S

U S

U S

U S

U S

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

P a y o ffs1 ,5 ,5

5 ,5 ,2

5 ,2 ,5

3 ,4 ,4

2 ,5 ,5

4 ,3 ,4

4 ,4 ,3

4 ,4 ,4

D ell

H P

H P

G atew ay

G atew ay

G atew ay

G atew ay

U S

U S

U S

U S

U S

U S

U S

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

P a y o ffs1 ,5 ,5

5 ,5 ,2

5 ,2 ,5

3 ,4 ,4

2 ,5 ,5

4 ,3 ,4

4 ,4 ,3

4 ,4 ,4

To prune the tree.

First consider all the choices Gateway would not make and strike out these branches.

Then prune out the choices HP would not make.

Finally prune out the choices Dell would not make.

D ell

H P

H P

G atew ay

G atew ay

G atew ay

G atew ay

U S

U S

U S

U S

U S

U S

U S

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

F o re ign

P a y o ffs1 ,5 ,5

5 ,5 ,2

5 ,2 ,5

3 ,4 ,4

2 ,5 ,5

4 ,3 ,4

4 ,4 ,3

4 ,4 ,4

Bad news for DellBad news for Dell

The Computer Location GameThe Computer Location Game


Sequential move gamesSequential move gamesMore than 2 movesMore than 2 moves

Suppose you are playing a game like chess.Suppose you are playing a game like chess.In principle the game can be solved by backwards In principle the game can be solved by backwards induction.induction.In practice it is too complicated.In practice it is too complicated.This does not mean you don’t think ahead.This does not mean you don’t think ahead.Think ahead a few moves and assess which situations Think ahead a few moves and assess which situations are possible.are possible.Give a value to each of these intermediate situations.Give a value to each of these intermediate situations.Play for highest intermediate value.Play for highest intermediate value.In chess this might involve creating space to maneuver In chess this might involve creating space to maneuver your pieces.your pieces.


Sequential move gamesSequential move gamesA Happy-Families example.A Happy-Families example.

Suppose Happy-Families and a rival are both developing similar Suppose Happy-Families and a rival are both developing similar new products. Each has to chose between an expensive high-tech new products. Each has to chose between an expensive high-tech variant and an inexpensive cruder version. Happy-Families have a variant and an inexpensive cruder version. Happy-Families have a lead in the technological race, and so will be able to bring their lead in the technological race, and so will be able to bring their product to market first. They must choose high or low tech before product to market first. They must choose high or low tech before their rival commits to their choice. The sequential nature of this their rival commits to their choice. The sequential nature of this game may be to Happy-Families' advantage or disadvantage.game may be to Happy-Families' advantage or disadvantage.

Advantage: Advantage: Getting to market first will allow customer loyalty to Getting to market first will allow customer loyalty to develop.develop.

Disadvantage: Disadvantage: The rival may observe market response to Happy-The rival may observe market response to Happy-Families’ product and use this information to choose a different Families’ product and use this information to choose a different expense/technology combination for their product.expense/technology combination for their product.

games people play. 2. sequential move games. in this section we shall learn how to figure out what...

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sequential games

ads gary

game treesa game tree

gubernatorial race game

sequential moves

happyfamilies nodes

possible payoffs

potential payoffs