Dynamic Games

Overview

In this unit we study: Combinations of sequential and simultaneous

games Solutions to these types of games Repeated games How to use dynamics to build self-sustaining

agreements

Sequential and Simultaneous Games

There are many situations where the strategic situation has both simultaneous and sequential elements

Examples: Decision by a firm to enter a market followed

by competition in pricing and advertising Decisions by candidates to run for office

followed by voting Attempts at legal settlements followed by trial

in the event no settlement is reached

How to Analyze these Games?

In sequential games, we saw that it paid to look forward and reason back Find the best decision a player can make on reaching

a point in the game “Prune” the game tree to eliminate worse (dominated)

decisions In simultaneous games, we looked for a best

response to a best response (Nash equilibrium). Project the strategy of a rival Choose a best response to that strategy Check if the rival would want to change his “projected”

strategy in view of your move.

New Elements to Dynamic Games

History matters Strategy is now based not only on projections

of the future and the present but also on the past.

The process by which you arrived at a point in the game might matter

Alternative futures matter Histories of the game that no one

contemplated as arising can play a key role in influencing outcomes

Entry

Consider the following situation: An incumbent firm is presently operating in some

profitable market A potential rival firm is considering entering this market Upon entry, the rival and the incumbent simultaneously

decide whether to fight or not fight If they both fight, then the profitability of the market is

such that the rival would be better off staying out If they do not fight, the rival would be better off entering

Game Tree

rival

Don’t fight

Fight

Don’t fight

a, a d, c

Fight c, d b, b

2a, e

out

in

Incumbent is row player.

Payoffs are (incumbent, rival)

Description

One interpretation of this diagram is the following: There are three levels of profitability in the market:

High (when no one fights) Medium (when 1 firm fights) Low (when both fight)

The product of fighting is to capture market share from the rival

If both fight or both don’t fight, market share is 50-50 If one fights and the other doesn’t the fighter gains

market share at the expense of the rival

Goal

Your goal is to provide an analysis of under what conditions to enter this market.

What are the key things to think about in making this decision?

Example 1: Large Market Share Capture Suppose that the profitability of the market is:

Big (No one fights)= 32 Medium (One firm fights)= 25 Small (Both fight) = 16

The outside option of a rival who does not enter is 11

When 1 firm fights and the other does not, the fighter obtains 80% market share

Game Tree: Example 1

rival

Don’t fight

Fight

Don’t fight

16, 16 5, 20

Fight 20, 5 8, 8

32, 11

out

in

Analysis

If the rival enters: Incumbent’s best response to don’t fight is to

fight Incumbent’s best response to fight is to fight The situation is symmetric for the rival

Therefore: If enter, then (fight, fight)

Game Tree: Example 1

rival

Don’t fight

Fight

Don’t fight

16, 16 5, 20

Fight 20, 5 8, 8

32, 11

out

in

Since the rival anticipates a fight on entering, it is better not to enter

Game Tree: Example 1- Generalized

rival

Don’t fight

Fight

Don’t fight

a, a d, c

Fight c, d b, b

2a, e

out

in

a > e > b (It only pays to enter absent a fight)

1. c > a

2. b > d

Game Tree: Example 1- Generalized

rival

Don’t fight

Fight

Don’t fight

a, a d, c

Fight c, d b, b

2a, e

out

in

a > e > b (It only pays to enter absent a fight)

1. c > a

2. b > d

Notice that the game after entry is a Prisoner’s dilemma

In this case the incumbent uses the b, b outcome to successfully deter entry

Comments

Notice that what didn’t happen –entry – had a profound effect on what did

The rival could count on the fact that entry combined with the temptation to grab market share would lead to a fight

Therefore, it paid to stay out of this market.

Example 2: Smaller Market Share Grab Suppose that the market sizes are again

Big = 32 Medium = 25 Small = 16

The outside option of a rival who does not enter is still 11

When 1 firm fights and the other does not, the fighter obtains 60% market share

Now what happens?

Game Tree: Example 2

rival

Don’t fight

Fight

Don’t fight

16, 16 10, 15

Fight 15, 10 8, 8

32, 11

out

in

Analysis

If rival enters: Incumbent’s best response to don’t fight is don’t fight Incumbent’s best response to fight is don’t fight Same for rival So neither fight if rival enters

If rival does not enter Incumbent is free to threaten to do whatever it likes In particular, it can threaten to fight In which case it pays for the rival to stay out

Equilibria

1. Rival enters, neither firm fights

2. Rival doesn’t enter, incumbent threatens to fight if it did enter

Notice that now entry deterrence depends crucially on the rival’s beliefs about the incumbent’s response

If the rival is convinced that the incumbent will be aggressive, it should not enter

Since the rival chooses not to enter, choosing to actually be aggressive is a best response by incumbent

Game Tree: Example 2 Generalized

rival

Don’t fight

Fight

Don’t fight

a, a d, c

Fight c, d b, b

2a, e

out

in

a > e > b

1. a > c

2. d > b

Game Tree: Example 2 Generalized

rival

Don’t fight

Fight

Don’t fight

a, a d, c

Fight c, d b, b

2a, e

out

in

a > e > b1. a > c2. d > b

Even though it is a dominant strategy for incumbent to not fight

It can deter entry by threatening.

Since in the even of successful deterrence, the threat is no tested this is still a best response for the incumbent

Comments

So it would seem that if the incumbent can affect the rival’s beliefs, it is possible to deter entry even in this framework.

Choosing Equilibria

The prospect that a threat which is costly to carry out might succeed in a situation like this posed a problem for game theory

Is there some rational means to choose between the equilibria?

Subgame Perfect Equilibria

We’ll generalize the idea of look forward, reason back the following way:

Rationality Axiom 2: When presented with any history of the game (even an unexpected one), players should choose best responses to future beliefs

Formally, we require that players choose optimizing strategies everywhere in the game

Example 2: Refined

Recall that not fighting was a dominant strategy for each of the players if the rival enters

Therefore, despite incumbent’s threats to the contrary Rival should anticipate that its entry will not

lead to fighting Therefore, it pays to enter.

Entry deterrence is not credible in this case.

Example 3: Shrinking Markets

Now consider a variation of example 2. Suppose that when either firm fights, it still

gains 60% market share but the profitability of the market shrinks to a greater extent than before.

Does this change rival’s view of the incumbent’s threat to fight?

Example 3: Specifics

Suppose that the market sizes are now Big = 32 Medium = 18 Small = 16

The outside option of a rival who does not enter is 11

When 1 firm fights and the other does not, the fighter still obtains 60% market share

Game Tree: Example 3

rival

Don’t fight

Fight

Don’t fight

16, 16 7.2,10.8

Fight 10.8, 7.2

8, 8

32, 11

out

in

Analysis

The best response to not fighting is not to fight The best response to fighting is to fight Therefore, there are 2 equilibria following entry If rival anticipates a fight, it should not enter If it anticipates no fighting, it should Hence there are 2 equilibria of the dynamic game:

Out -> Fight, Fight In -> Don’t fight, Don’t fight

Comments

Notice that making competition more disruptive on entry actually improves the credibility of the threat by the incumbent

In many situations it is possible to control how destructive competition will be in markets

The subtlety here is that it’s in the incumbent’s interest to make the destructiveness of competition more rather than less

Game Tree: Example 3 Generalized

rival

Don’t fight

Fight

Don’t fight a, a d, c

Fight c, d b, b

2a, e

out

in

a > e > b

1. a > c

2. b > d

Game Tree: Example 2 Generalized

rival

Don’t fight

Fight

Don’t fight

a, a d, c

Fight c, d b, b

2a, e

out

in

a > e > b1. a > c2. b > d

Now this is a coordination game if the rival enters.

Either both firms can coordinate on not fighting, or they can coordinate on fighting

Either is self-sustaining and the fighting outcome deters entry.

Key Conclusions

The idea of subgame perfection is to assess the credibility of threats. We’ll return to this issue in the next class

Threats which are not self-sustaining if carried out, should correctly be viewed with skepticism

To make a threat credible, it can sometimes serve the interest of the incumbent to destroy profitability of the market in the event of entry.

Repeated Games

We turn now to repeated games These are games where players are involved

in the same (or similar) strategic situation for many periods in a row.

The key insight here will be that we can use the future to affect the outcome in the present.

Example 1 Revisited

Suppose that entry has occurred and that the situation is as in example 1

Example 1: Game Table

Suppose a > b

Competition is destructive to profitability

c > a Market share grabs

are profitable b > d

Fighting back is better than being a victim of a grab

Don’t fight

Fight

Don’t fight

a, a d, c

Fight c, d b, b

Analysis

We determined that fighting was the inevitable outcome With consequent decrease in the profitability of the

market Suppose that the firms will compete for 2 periods

instead of 1? Firm 1 and 2 agree to the following:

Don’t fight in either period If either of us fights in the first period, fight in the

second Will this work?

Carrots and Sticks

Each firm is holding out a carrot—the promise of a in both periods

And a stick, the threat of b in the second period

To try to deter the temptation to grab market share (and get c)

This could work if 2a > c + b

Look Forward Reason Back

But there’s a problem here: In period 2, there’s no stick and no carrot So each firm will be tempted to fight and

succumb to that temptation And this “reverberates” back to period 1

Each firm knows that there is no “carrot” in the second period---only the “stick”

So there’s nothing to deter the temptation to fight in period 1

Hence the firms will fight in both periods

Generalizing

The principle applies for any set number of periods.

Since there’s nothing to promote good behavior at the end of the game, firms will fight then And this reverberates backward throughout

the game The conclusion is a sad one:

If the game has any set ending time, the firms will fight in every period

Infinite Repetition

Suppose there is no fixed endpoint to the game

Instead the firms expect the game to be infinitely repeated This is an abstraction---think of it as a game

being repeated for a really long time with no one knowing exactly when it will end

Now can the firms cooperate?

Tit-for-Tat Strategies

Suppose the firms make the following deal: We agree not to fight If my rival fights, I’ll fight in the next period Then the war is over and we’ll resume not fighting

Will this work? Now in every period the firm must weigh the gains from

cheating (i.e. the temptation) c – a

Versus the cost in the future (the carrot and stick) a – b

If the cost exceeds the temptation, each firm will refrain from fighting.

a – b > c – a

Variations

Suppose the carrot and stick in the above agreement are not enough to overcome the temptation i.e., a – b < c – a

Is all lost? No. Because there is always a future, the size

of the “stick” can be increased

Tit-for-2 Tats

Suppose the firms make the following deal: We agree not to fight If my rival fights, I’ll fight for the next 2 periods Then the war is over and we’ll resume not fighting

Will this work? Now in every period the firm must weigh the gains from

cheating (i.e. the temptation) c – a

Versus the cost in the future (the carrot and stick) 2 x (a – b)

If the cost exceeds the temptation, each firm will refrain from fighting.

2 x (a – b) > c – a

Key Point

The larger is the temptation or the weaker the punishment available in any one period

The longer the threat of “war” needs to be to deter cheating.

The size of the stick needs to be calibrated to the upside from cheating.

Obviously the promise of infinite war leads to the largest possible “stick”

Discounting

All of this assumed that profits in the future were worth the same as those in the present

Of course, they’re not Suppose that we discount profits in the future

by the real interest rate r. How does this change the analysis?

Tit for n Tats

If we play tit-for-tat: NPV of temptation = c – a NPV of the threat = (a – b)/(1 + r)

Notice that as the interest rate increases, the punishment associated with the threat declines

Perpetual Punishment

In a world with positive real interest rates, even the threat of perpetual punishment may not be enough to stave off fighting

Recall the perpetuity formula: What is the NPV of an asset has a cash flow C

starting 1 period in the future and lasting in perpetuity?

NPV = C/r

Analysis

Suppose we threaten punishment forever after a defection.

NPV of Temptation = c – a NPV of punishment = (a – b)/r

For higher real interest rates, even the most severe possible punishment loses efficacy

If this threat is insufficient to deter cheating, nothing will deter it.

More Variations to Consider

It is enough that there is a possibility that the game will continue into the future to use threat to sustain cooperation

Like the interest rate, the higher the chance the game will end, the less powerful the threat and the harder to sustain agreement

Like the entry games, threat need to be credible---a strong punishment strategy that is not credible to implement is not an effective deterrent.

Using Threats to Sustain Cooperation What makes for a good threat?

Detection Clarity Repetition and reputation Credibility of enforcement

What about forgiveness? What about errors?

Conclusion

This suggests: Severe but credible punishments Limited temptation from short run abuses Strong upside incentives to cooperate.

If costs are increasing in the severity of each measure then a mix of measures is best.