4.2 — Credible Commitments

# 4.2 — Credible Commitments
## ECON 316 • Game Theory • Fall 2021
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/gameF21"><i class="fa fa-github fa-fw"></i>ryansafner/gameF21</a><br> <a href="https://gameF21.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>gameF21.classes.ryansafner.com</a><br>

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# Outline

### [Credible Commitments](#3)
### [Threats and Applications](#9)
### [Promises and Applications](#27)

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# Credible Commitments

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# Another Motivating Example: Why Professors Are Mean

.pull-left[
.smallest[
- Most professors have a lateness policy where late homework is either not accepted, or points are lost

- Not (necessarily) because professors are mean!

- Suppose a student hands in homework late and has a plausible excuse

- Most professors actually are generous and accommodating, will make an exception

- But if students know this, all students will try plausible excuses and everything becomes late
]
]

---

# Another Motivating Example: Why Professors Are Mean

.pull-left[
.smaller[
- Professor can **commit** to a bright-line policy from the beginning (i.e. in syllabus)

- Removes professor's discretion in individual cases

- The *policy* may be "mean", but leads to a better Nash equilibrium by tying professor's hands

- Salespeople have same limitations from "manager" or "man upstairs" preventing better deals
]
]

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# What Doesn't Kill You Makes You Stronger

- **Committing** to something is costly in the short-run, but often makes the commit-er better off in the long run

- Often need some kind of **commitment device** to artificially constrain your ability to react

]

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# What Doesn't Kill You Makes You Stronger

1921—2016

Economics Nobel 2005
]
]
]

> “Bargaining power"...suggest[s] that the advantage goes to the powerful, the strong, or the skillful. It does, of course, if those qualities are defined to mean only that negotiations are won by those who win...The sophisticated negotiator may find it difficult to seem as obstinate as a truly obstinate man,” (p.22).

> “.hi[Bargaining power [is] the power to bind oneself],” (p.22).

]
.source[Schelling, Thomas, 1960, *The Strategy of Conflict*]
]

---

# What Doesn't Kill You Makes You Stronger

1921—2016

Economics Nobel 2005
]
]
]

> “.hi[How can one commit himself in advance to an act that he would in fact prefer not to carry out in the event,] in order that his commitment may deter the other party? ... In bargaining, the commitment is a device to leave the last clear chance to decide the outcome with the other party, in a manner that he fully appreciates; it is to relinquish further initative, having rigged the incentives so that the other party must choose in one's favor. If one driver speeds up so that he cannot stop, and the other realizes it, the latter has to yield...This doctrine helps to understand some of those cases in which .hi[bargaining 'strength' inheres in what is weakness by other standards.],” (p.22).

]
.source[Schelling, Thomas, 1960, *The Strategy of Conflict*]
]

---

# Why Are the Following So Difficult?

- Waking up early

- Dieting

- Going to the gym
]

![:scale 80%](../images/timeinconsistency.png)

]
]

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# Time-inconsistency Problem

- .hi[Time inconsistency problem]: *Future you* will have different preferences at the moment of truth than *Present you* has now
]

![:scale 80%](../images/timeinconsistency.png)

]
]

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# Time Inconsistency and Commitment Devices

.pull-left[
.smallest[
- With a .hi[commitment device] you can bind yourself in the future to obey your present wishes

- Limiting your future choices keeps your preferences consistent over time

- Examples:
  - Deadlines
  - Rely on other people
  - Stake your reputation on it
  - Impose a high cost on yourself for failure
  - Hire an agent who is compensated based on your success
]
]

![:scale 75%](../images/alarmshredder.jpg)

]
]

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# Ways to Commit and Make Strategies Credible

.pull-left[
.smallest[
- *Dixit and Nalebuff* (Ch. 7) describe 8 methods to make strategies credible (and also suggestions for countering them):

1. Write enforceable contracts
2. Establish and stake your reputation on your actions
3. Cut off communication
4. Burn bridges behind you
5. Leave the outcome beyond your control, or to chance
6. Move in small steps
7. Develop credibility through teamwork
8. Employ mandated agents
]
]

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# Write Contracts

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# Write Contracts

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# Write Contracts

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# Cut Off Communication

- Take the Assurance game example

- Suppose .red[Harry] publicly announces “I'm going to Whitaker” and then walks away (and turns off his phone), unable to be reached
]

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# Cut Off Communication

- Take the Assurance game example

- Suppose .red[Harry] publicly announces “I'm going to Whitaker” and then walks away (and turns off his phone), unable to be reached

- If .blue[Sally] believes him, she has little choice but to go to .blue[Whitaker]
]

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# Cut off Communication

.center[
<iframe width="980" height="550" src="https://www.youtube.com/embed/PSofqNSuVy8" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
]

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# Stake Your Reputation on Performance

.smaller[
> “Given the importance of credit sales, the diamond industry depends overwhelmingly on the reliable enforcement of executory contracts. However, while most industries employ state-sponsored courts to enforce payment after the delivery of goods, public courts are toothless to enforce credit sales for diamonds. Diamons are easily portable and command extreme value throughout the world. A diamond thief encounters little difficulty in hiding unpaid-for or stolen diamonds from law enforcement officials, fleeing American jurisdiction, and selling the valuable diamonds to black market buyers,” (p.392).

]

.source[Richman, Barak D, 2006, “How Community Institutions Create Economic Advantage: Jewish Diamond Merchants in New York,” *Law and Social Inquiry* 31(2):383-420]
]

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# Stake Your Reputation on Performance

> “The failure of public courts requires diamond merchants to rely on trust-based exchange. Mutual trust among merchants -- which the *New York Times* has called "the real treasure of 47th street" -- assures dealers that by maintaining a trustworthy reputation, they will remain in good community standing and preserve the opportunity to engage in future lucrative transactions...despite the unreliability of state courts.,” (p.393).

.source[Richman, Barak D, 2006, “How Community Institutions Create Economic Advantage: Jewish Diamond Merchants in New York,” *Law and Social Inquiry* 31(2):383-420]
]

---

# Burn Your Boats

- .blue[Hernan Cortes] and the .blue[Spanish conquistadors] invade Mexico in the early 16<sup>th</sup> century, ruled by .red[Aztecs]

- If both sides fight, worst outcome for both
  - .blue[Spaniards] have inferior numbers than .red[Aztecs], heavier losses

- If one side fights and the other runs, the fighter gets more than runner

- If both sides run, nothing happens

]

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# Burn Your Boats

- .blue[Hernan Cortes] and the .blue[Spanish conquistadors] invade Mexico in the early 16<sup>th</sup> century, ruled by .red[Aztecs]

- If both sides fight, worst outcome for both
  - .blue[Spaniards] have inferior numbers than .red[Aztecs], heavier losses

- If one side fights and the other runs, the fighter gets more than runner

- If both sides run, nothing happens

]

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# Burn Your Boats

- .hi-purple[SNPE]: {.red[Fight], .blue[(Run, Fight)]}

- Spaniards lose
  - and no credible threat to respond to .red[Fight] with .blue[Fight]

]

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# Burn Your Boats

- Cortes decides before the game begins to make a .hi[strategic move]: burn his ships so his men cannot retreat

- Removes option of .blue[Run] for .blue[Spaniards]

]

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# Burn Your Boats

- Cortes decides before the game begins to make a .hi[strategic move]: burn his ships so his men cannot retreat

- Removes option of .blue[Run] for .blue[Spaniards]

- Now resolve for .hi-purple[SPNE]

]

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# Burn Your Boats

- .hi-purple[SPNE]: {.red[(Run, Fight)], .blue[(Burn, Fight, Fight, Run, Fight)]}

- .blue[Spaniards]’ pre-game strategic move of .blue[Burn] set them up for a superior outcome (for them): .blue[Burn] `$\rightarrow$` .red[Run] `$\rightarrow$` .blue[Fight]
]

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# This Is a Classic Military Tactic

> “If you want to take the island, burn the boats.” — Julius Caesar before his invasion of Britannia (55 B.C.)

]

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# This Is a Classic Military Tactic

> “At the critical moment, the leader of an army acts like one who has climbed up a height, and then kicks away the ladder behind him” — Sun Tzu, *The Art of War* (c.500 B.C.)

]

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# Take the Result out of Your Hands

1921—2016

Economics Nobel 2005
]
]
]

> “Making a credible threat involves proving that one would have to carry out the threat, or creating incentives for oneself or incurring penalties that would make one evidently want to. The acknowledged purpose of stationing American troops in Europe as a “trip wire” was to convince the Russians that war in Europe would involve the United States whether the Russians thought the United States wanted to be involved or not -- that escape from the commitment was physically impossible.” (p.187).

]
.source[Schelling, Thomas, 1960, *The Strategy of Conflict*]
]

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# Take the Result out of Your Hands

1921—2016

Economics Nobel 2005
]
]
]

> “The key to these threats is that, though one party may or may not carry them out if the threatened party fails to comply, .hi[the final decision is not altogether under the threatener's control],” (p.187).

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# Take the Result out of Your Hands

1921—2016

Economics Nobel 2005
]
]
]

.smallest[
> “Ideally, for this purpose, I should have a little black box that contains a roulette wheel and a device that will detonate in a way that unquestionably provokes total war...I tell [the Russians]—.hi[demonstrate to them]—that the little box will keep running until my demands have been complied with and that .hi[there is nothing I can do to stop it]...Note that I do not insist that I shall .hi[decide] on total war...I leave it all up to the box which .hi[automatically] engulfs us both in war if the right (wrong) combination comes up on any day.” (p.197).

]

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# Take the Result out of Your Hands

1921—2016

Economics Nobel 2005
]
]
]

.smallest[
> “Brinkmanship is thus the deliberate creation of a recognizable risk of war, a risk that one does not completely control. It is the tactic of deliberately letting the situation get somewhat out of hand, just because its being out of hand may be intolerable to the other party and force his accomodation. It means harassing and intimidating an adversary by exposing him to a shared risk, or deterring him by showing that if he makes a contrary move he may disturb us so that we slip over the brink whether we want to or not, carrying him with us,” (p.200).

]

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# The Doomsday Device

.center[
<iframe width="980" height="550" src="https://www.youtube.com/embed/2yfXgu37iyI" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
]

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# Threats and Applications

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# Threats: Entry Game Example

- Consider our .hi[Entry Game], between a potential .hi-red[Entrant] and an .hi-blue[Incumbent], from before
]

]

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# Threats: Entry Game Example

- Consider our .hi[Entry Game], between a potential .hi-red[Entrant] and an .hi-blue[Incumbent], from before
]

]

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# Threats: Entry Game Example

- Two Nash Equilibria:
1. (.hi-red[Enter], .hi-blue[Accommodate])
2. (.hi-red[Stay Out], .hi-blue[Fight])

]

![:scale 80%](../images/entry_game_normal_NE.png)

]

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# Threats: Entry Game Example

- Two Nash Equilibria:
1. (.hi-red[Enter], .hi-blue[Accommodate])
2. (.hi-red[Stay Out], .hi-blue[Fight])

- Only (.hi-red[Enter], .hi-blue[Accommodate]) is a .hi-purple[Subgame Perfect Nash Equilibrium (SPNE)]

- These strategy profiles for each player constitute a Nash equilibrium in every possible subgame!
]

![:scale 80%](../images/entry_game_normal_NE.png)

]
]

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# Threats: Entry Game Example

- Suppose before the game started, .hi-blue[Incumbent] announced to .hi-red[Entrant]

> “if you .red[Enter], I will .blue[Fight]!”

- This **threat** is .hi-purple[not credible] because playing .blue[Fight] in response to .red[Enter] is not rational!

- The strategy is not Subgame Perfect!
]

![:scale 80%](../images/entry_game_normal_NE.png)

]
]

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# Threats: Entry Game Example with Commitment

- Suppose before the game started, .hi-blue[Incumbent] could decide whether or not to .blue[Invest] in excess capacity

- This is costly, suppose .blue[Incumbent] incurs a cost of .blue[-1]

- Builds up a “war chest” allowing .blue[Incumbent] to survive a price war

- Now suppose .blue[Incumbent] makes same threat to .red[Entrant]:

> “if you .red[Enter], I will .blue[Fight]!”
]

![:scale 70%](../images/cableconstruction.jpg)

]
]

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# Threats: Entry Game Example with Commitment

- Game changes, a first stage where .hi-blue[Incumbent] goes first at (new) .blue[I.1], deciding whether to .blue[Invest] or .blue[Don't]
  - Game is the same as before from .red[E.2] onwards
  
- This is a more complicated game, let's apply what we've learned about subgame perfection...

]

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# Threats: Entry Game Example with Commitment

- What are the subgames?

]

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# Threats: Entry Game Example with Commitment

- What are the subgames?

1. Subgame initiated by node .blue[I.1] (game itself)
2. Subgame initiated by node .red[E.1] 
3. Subgame initiated by node .red[E.2]
4. Subgame initiated by node .blue[I.2] 
5. Subgame initiated by node .blue[I.3]

]

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# Threats: Entry Game Example with Commitment

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# Threats: Entry Game Example with Commitment

- .hi-red[Entrant], choosing at nodes .red[(E.1, E.2)]
  1. .red[(Stay Out, Stay Out)]
  2. .red[(Stay Out, Enter)]
  3. .red[(Enter, Stay Out)]
  4. .red[(Enter, Enter)]
]
]

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# Threats: Entry Game Example with Commitment

- .hi-blue[Incumbent], choosing at nodes .blue[(I.1, I.2, I.3)]
  1. .blue[(Invest, Accommodate, Accommodate)]
  2. .blue[(Invest, Accommodate, Fight)]
  3. .blue[(Invest, Fight, Accommodate)]
  4. .blue[(Invest, Fight, Fight)]
  5. .blue[(Don't, Accommodate, Accommodate)]
  6. .blue[(Don't, Accommodate, Fight)]
  7. .blue[(Don't, Fight, Accommodate)]
  8. .blue[(Don't, Fight, Fight)]
]
]

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# Threats: Entry Game Example with Commitment

1. {.red[(O,O)], .blue[(D,A,F)]}
2. {.red[(O,O)], .blue[(D,F,F)]}
3. {.red[(O,E)], .blue[(I,F,A)]}
4. {.red[(O,E)], .blue[(I,F,F)]}
5. {.red[(O,E)], .blue[(D,A,A)]}
6. {.red[(O,E)], .blue[(D,F,A)]}
7. {.red[(E,O)], .blue[(D,A,F)]}
8. {.red[(E,O)], .blue[(D,F,F)]}
9. {.red[(E,E)], .blue[(D,A,A)]}
10. {.red[(E,E)], .blue[(D,F,A)]}

...which is .hi-purple[subgame perfect]?
]
]

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# Threats: Entry Game Example with Commitment

- Solve the game in sequential form via backward induction
]
.pull-right[
.center[
![](../images/entry_game_commitment.png)
]
]

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# Threats: Entry Game Example with Commitment

- Solve the game in sequential form via backward induction

- .hi-purple[SPNE]: {.red[(O,E)], .blue[(I,F,A)]}
]
.pull-right[
.center[
![](../images/entry_game_commitment_rollback.png)
]
]

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# Threats: Entry Game Example with Commitment

.center[
![](../images/entry_game_commitment_rollback.png)
]
]
.pull-right[
.center[
![](../images/entry_game_commitment_normal_SPNE.png)
]
]

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# Threats: Entry Game Example with Commitment

- .hi-purple[SPNE]: {.red[(O,E)], .blue[(I,F,A)]}

- This set of strategies induces a Nash equilibrium in all (5) subgames
]
.pull-right[
.center[
![](../images/entry_game_commitment_SPNE.png)
]
]

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# Threats: Entry Game Example with Commitment

- Recall .hi-blue[Incumbent]’s threat to .hi-red[Entrant]

> “if you .red[Enter], I will .blue[Fight]!”

- With **commitment**, it is .hi-purple[credible] for .blue[Incumbent] to threaten to .blue[Fight] if .red[Entrant] decides to .red[Enter]!

]
.pull-right[
.center[
![](../images/entry_game_commitment_SPNE.png)
]
]

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# Why The U.S. Hasn’t Bombed North Korea

- Why hasn’t the U.S. bombed North Korea?

]

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# Why The U.S. Hasn’t Bombed North Korea

- Why hasn’t the U.S. bombed North Korea?

]

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# Why The U.S. Hasn’t Bombed North Korea

]

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# Why The U.S. Hasn’t Bombed North Korea

- Why hasn’t the U.S. bombed North Korea?

- Suppose placing and constantly hiding artillery costs .blue[North Korea -3]

]

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# Why The U.S. Hasn’t Bombed North Korea

- Why hasn’t the U.S. bombed North Korea?

- Suppose placing and constantly hiding artillery costs .blue[North Korea -3]

- A credible threat to .blue[Bomb Seoul] in response to a .red[U.S. Attack]
]

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# Promises and Applications

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# Promises

- .red[Principal] decides to .red[invest] money ($100) with .blue[Agent]
  - Investment grows to $200

- .blue[Agent] can then .blue[keep] or .blue[share] the returns with .red[Principal]
]
]
.pull-right[
.center[
![:scale 75%](../images/pa_game.png)

![:scale 80%](../images/pa_game_normal.png)
]
]

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# Promises

- Only one Nash equilibrium, which is .hi-purple[SP]: {.red[Don't], .blue[Keep]}
]
]
.pull-right[
.center[
![:scale 75%](../images/pa_game_rollback.png)

![:scale 80%](../images/pa_game_normal_NE.png)
]
]

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# Promises

- Only one Nash equilibrium, which is .hi-purple[SP]: {.red[Don't], .blue[Keep]}

- What if before game began, .blue[Agent] said to .red[Principal]:

> “If you .red[Invest], I will .blue[Share]”

]
]
.pull-right[
.center[
![:scale 75%](../images/pa_game_rollback.png)

![:scale 80%](../images/pa_game_normal_NE.png)
]
]

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# Promises

- Only one Nash equilibrium, which is .hi-purple[SP]: {.red[Don't], .blue[Keep]}

- What if before game began, .blue[Agent] said to .red[Principal]:

> “If you .red[Invest], I will .blue[Share]”

- Not a credible promise, not subgame perfect!
]
]
.pull-right[
.center[
![:scale 75%](../images/pa_game_rollback.png)

![:scale 80%](../images/pa_game_normal_NE.png)
]
]

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# Making Promises Credible

.pull-left[
.smallest[
- One solution: .hi-purple[reputation], which acts like a forfeitable .hi-purple[bond]

- If .blue[Agent] chooses to .blue[Keep], will lose -H, which is “hostage” value
  - .red[Principal] will earn `$\alpha H$`, where `$\alpha$` is the faction of `$H$` that is valuable to .red[Principal]
  - `$\alpha = 0$`: hostage has no value to .red[Principal]
  - `$\alpha = 1$`: cash

]
]

]

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# Making Promises Credible

.pull-left[
.smallest[
- One solution: .hi-purple[reputation], which acts like a forfeitable .hi-purple[bond]

- If `$H >150$` and `$\alpha H > 100$`, .hi-purple[SPNE]: (.red[Invest, Don't], .blue[Bond, Share, Keep])

]
]

]

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# Making Promises Credible

.pull-left[
.smallest[
- Common type of bond is reputation, which people can invest in and is “held hostage” for good behavior
  - reneging on commitments destroys reputation
  - works best with repeat interactions, high discount rates (folk theorem!)
  
- Another is some collateral property that is forfeit if the contract is breached
  - Mortgages, secured loans, etc

]
]

]

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# In The Old Days, These Were *Actual* Hostages

.source[Williamson, Oliver E, 1983, “Credible Commitments: Using Hostages to Support Exchange,” *American Economic Review* 73(4): 519–540]

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# Today We Often Hold Property Hostage as Collateral

.source[Williamson, Oliver E, 1983, “Credible Commitments: Using Hostages to Support Exchange,” *American Economic Review* 73(4): 519–540]

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# Contract Law: Making Promises Credible

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# Contract Law: Making Promises Credible

.pull-left[
- Suppose instead we have courts enforce a promise to .blue[Keep]
  - Court will force .blue[Agent] to give $150 to .red[Principal]
  - Litigation cost of using courts `$c$` to each party
  
- With `$c>0$`, .hi-purple[SPNE]: (.red[Invest], .blue[Share])
  - (One main) purpose of contract law is to .hi-purple[make promises credible]
]

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# Making Promises Credible: Engagement