UNIT I:Overview & History Introduction: What is Game Theory? Von Neumann and the Bomb The Science of...

UNIT I: Overview & History

• Introduction: What is Game Theory?• Von Neumann and the Bomb• The Science of International Strategy• Logic of Indeterminate Situations

2/13

The Science of International Strategy

• Schelling’s Reorientation of Game Theory• Securing Insecure Agreements• Post War Economic Regimes• Bargaining and Negotiation

Last time, we considered the application of game theory to problems of military strategy (e.g., Battle of Bismarck Sea; “Dr. Strangelove”).

Today, we will consider game theory’s role in analyzing the US-Soviet conflict, in particular the move from zero- to nonzero-sum game theory: from a theory of war to a theory of bargaining and negotiation.

The Science of International Strategy

Realism

• The actor (nation-state) is rational: goal-directed, concerned with maximizing power or security.

• The environment is anarchic: : there is no supervening authority that can enforce agreements.

• The solution is an equilibrium or balance of power, enforced by the interests of those involved w/o the need for external enforcement mechanisms.

Schelling’s Reorientation

• In the 1940s and ’50s, game theory lent itself to the analysis of military strategy, casting Realist principles and assumptions at an abstract level of analysis.

• Von Neumann’s minimax theorem and the doctrine of military decision both recommend prudence: maximize the minimum payoff available.

• Given Realist assumption, conflict is inevitable. The Security Dilemma arises because one nation’s attempt to increase it’s security decreases the security of others.

• Arm Races (e.g., WWI). Is security zero-sum?


The Security Dilemma• The actor (nation-state) is rational, i.e., goal-directed,

egoistic, concerned with maximizing power or security.

• The structure of the international system is anarchic – meaning there is no supervening authority that can enforce agreements.

• Given these conditions, nations often fail to cooperate even in the face of common interests.

• The dilemma arises because one nation’s attempt to increase it’s security decreases the security of others.


• In the 1940s and ’50s, game theory lent itself to the analysis of military strategy, casting Realist principles and assumptions at an abstract level of analysis.

• Von Neumann’s minimax theorem and the doctrine of military decision both recommend prudence: maximize the minimum payoff available.

• Given Realist assumption, conflict is inevitable. The Security Dilemma arises because one nation’s attempt to increase it’s security decreases the security of others.

• Arm Races (e.g., WWI). Is security zero-sum?


The Reciprocal Fear of Surprise Attack

The technology of nuclear warfare created a fundamentally new kind of arms race – the speed and devastation of the new generation of weapons meant that “[f]or the first time in the history of the world, it became possible to contemplate a surprise attack that would wipe the enemy off the face of the earth ... . Equally important, each nation would fear being the victim of the other’s surprise attack” (Poundstone, 1992, p. 4).


The Prisoner’s DilemmaThe prisoner’s dilemma is a universal concept. Theorists now realize that prisoner’s dilemmas occur in biology, psychology, sociology, economics, and law. The prisoner’s dilemma is apt to turn up anywhere a conflict of interests exists (..) . Study of the prisoner’s dilemma has great power for explaining why animal and human societies are organized as they are. It is one of the great ideas of the twentieth century, simple enough for anyone to grasp and of fundamental importance (...). The prisoner’s dilemma has become one of the premier philosophical and scientific issues of our time. It is tied to our very survival (Poundstone,1992: 9).

The Prisoner’s Dilemma

In years in jail Al

Confess Don’t

Confess

Bob

Don’t

10, 10 0, 20

20, 0 1, 1

Bob thinks:

If Al C(onfesses), I should C, because 10 < 20 and 0 < 1, thus C is better than D(on’t), no matter whatAl does.

We call Confess adominant strategy.


In years in jail Al

Confess Don’t

Confess

Bob

Don’t

10, 10 0, 20

20, 0 1, 1

Because the game is symmetric, both prisoner’s Confess, even though they are better off if both Don’t.

CC is inefficient.

If we assign P(ayoffs),so that the players try to maximize P . . .


In Payoffs Al

Confess Don’t

Confess

Bob

Don’t

1, 1 5, 0

0, 5 3, 3Again, the outcome isinefficient.

If we assign P(ayoffs),so that the players try to maximize P . . .


Communication? We have assumed that there is no communication between the two prisoners. What would happen if they could communicate?

Repetition? In the Prisoner’s Dilemma, the two prisoners interact only once. What would happen if the interaction were repeated?

2- v. n-person Games? The Prisoner’s Dilemma is a two-person game, What would happen if there were many players?

Dominance Reasoning? Compelling as the reasoning is that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Is it really the most “rational” answer after all?

So why didn’t the US preemptively attack the USSR?

Securing Insecure Agreements

[H]ow to make agreements work when trust and good faith are lacking and there is no legal recourse for breach of contract (Schelling, 1960: 20).

• Hostage or spy exchange • Burning one’s bridges (& other strategic paradoxes)• Ulysses and the Sirens• The Doomsday Machine• The Balance of Terror

The Balance of Terror: MAD

There is a difference between a balance of terror in which either side can obliterate the other and one in which both sides can do it no matter who strikes first. It is not the “balance” – the sheer equality or symmetry of the situation – that constitutes mutual deterrence; it is the [self-enforcing] stability of the balance. The balance is stable when neither, in striking first, can destroy the other’s ability to strike back (Schelling, 1960: 232).

Credible second-strike capability: An equilibrium



Prisoner’s Dilemma Assurance Game

3, 3 0, 5

5, 0 1, 1

Don’t

Build weapons

Don’t Build weapons



3, 3 0, 5

5, 0 1, 1

Don’t

Attack

Don’t Attack

MAD: Create a credible second strike capability

How would this change the game?



3, 3 0, 2

2, 0 1, 1

3, 3 0, 5

5, 0 1, 1

In the PD, the pareto efficient outcome is individually inaccessible and collectively unstable.

in the AG, it is individually inaccessible but collectively stable, I.e., self-enforcing.

We learned to live with MAD. Both sides built and targeted missiles, and neither attacked. The stability that emerged between East and West made it possible for each side to turn its attention inward, on its own sphere of influence. In the West, focus shifted from (zero-sum and bipolar) security problems to (multipolar and nonzero-sum) problems of reorganizing and reviving the national economies that had been battered by the war. Accordingly, game theory turned from a theory of war to a theory of exchange.

Postwar Economic Regimes


Bretton Woods (1944)

World Bank International Monetary Fund (IMF) General Agreement on Tariffs

& Trade (GATT) (1947)

World Trade Organization (WTO)(1994)

• Multilateral trade liberalization• International business negotiations• Trade and environment• North/South conflict


Bargaining Games

Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or Surplus:

S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price)

b

0 50 100 150 200 250

s

i) b > s

If b and s are known to both players: How should the surplus be divided?

Surplus = 50

Mixed Motives: The Bargain

Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b = s, we say the price is fully determined, and there is no room for negotiation.


b

0 50 100 150 200 250

s

ii) b = s

Mixed Motives: The Bargain

Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b < s, there is nothing to gained from the exchange.


b

0 50 100 150 200 250

s

(iii) b < s No “zone of agreement”

What happens if information is incomplete?

Acquiring a Company

BUYER represents Company A (the Acquirer), which is currently considering make a tender offer to acquire Company T (the Target) from SELLER. BUYER and SELLER are going to be meeting to negotiate a price.

Company T is privately held, so its true value is known only to SELLER. Whatever the value, Company T is worth 50% more in the hands of the acquiring company, due to improved management and corporate synergies. BUYER only knows that its value is somewhere between 0 and 100 ($/share), with all values equally likely.

Source: M. Bazerman

Acquiring a Company

What offer should Buyer make?

Acquiring a Company

5

Source: Bazerman, 1992

9

1 0

4 4 7

27

18

45

123 BU MBA Students

$0 10-15 20-25 30-35 40-45 50-55 60-65 70-75 80-85 90-95Offers

Acquiring a Company

$0 10-15 20-25 30-35 40-45 50-55 60-65 70-75 80-85 90-95

5

Source: Bazerman, 1992

9

1 0

4 4 7

27

18

45

123 BU MBA Students

Similar results from

MIT Master’s Candidates

CPA; CEOs.

Offers

Acquiring a Company

There are many ways to depict the decision tree for this game. Buyer moves first and can make any Offer from to.

Let’s say Buyer offers $60.

O(ffer) = 0 60 100

Buyer

SellerAccept Reject

O – s = -60 30

EP(O) = - 15

The expected payoff of a $60 offer is a net loss of - $15.

Acquiring a Company

O(ffer) = 0 60 100

Buyer

SellerAccept Reject

Chance

s < 60 s > 60

s = 0 60s – O = -60 30

EP(O) = - 15

The expected payoff of a $60 offer is a net loss of - $15.

There are many ways to depict the decision tree for this game. Buyer moves first and can make any Offer from to.

Seller accepts if O > s.

Acquiring a Company OFFER VALUE ACCEPT OR VALUE GAIN OR

TO SELLER REJECT TO BUYER LOSS (O) (s) (3/2 s = b) (b - O) $60 $0 A $0 $-60

10 A 15 -45 20 A 30 -30 30 A 45 -15 40 A 60 0 50 A 75 15 60 R - - 70 R - -

Acquiring a CompanyThe key to the problem is the asymmetric information structure of the game. SELLER knows the true value of the company (s). BUYER knows only the upper and lower limits (0 < s < 100). Therefore, buyer must form an expectation on s (s').

BUYER also knows that the company is worth 50% more under the new management, i.e., b' = 3/2 s'. BUYER makes an offer (O). The expected payoff of the offer, EP(O), is the difference between the offer and the expected value of the company in the hands of BUYER:

EP(O) = b‘ – O = 3/2s‘ – O.

Acquiring a CompanyBUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted:

EP(O) = b‘ – O = 3/2s‘ – O.

O = s' + . Seller accepts if O > s.

Now consider this: Buyer has formed her expectation based on very little information. If Buyer offers O and Seller accepts, this considerably increases Buyer’s information, so she can now update her expectation on s.

How should Buyer update her expectation, conditioned on the new information that s < O?


EP(O) = b‘ – O = 3/2s‘ – O.


Let’s say BUYER offers $50. If SELLER accepts, BUYER knows that s cannot be greater than (or equal to) 50, that is: 0 < s < 50. Since all values are equally likely, s''/(s < O) = 25. The expected value of the company to BUYER (b'' = 3/2s'' = 37.50), which is less than the 50 she just offered to pay. (EP(O) = - 12.5.) When SELLER accepts, BUYER gets a sinking feeling in the pit of her stomach.

THE WINNER’S CURSE!


EP(O) = b‘ – O = 3/2s‘ – O.


Then for any accepted Offer (O), BUYER's expected value of s (s'') will be ½O (-). Since b'' = 3/2s '' and s'' = ½ O (-), b'' = ¾O (-). Thus:

EP(O) = - ¼O (-). EP is negative for all values of O.

THE WINNER’S CURSE!

Acquiring a Company

• The high level of uncertainty swamps the potential gains available, such that value is often left on the table, i.e., on average the outcome is inefficient.

• Under these particular conditions, BUYER should not make an offer.

• SELLER has an incentive to reveal some information to BUYER, because if BUYER can reduce the uncertainty, she may make an offer that leaves both players better off.

Summary


• Game theory had its earliest successes as a theory of war.

• Schelling (and others) argued that the advent of nuclear warfare fundamentally changed the game; most interesting international conflicts are mixed-motive or nonzero-sum games.

• In this context, game theory becomes a tool for conflict resolution as an alternative to war (bargaining & negotiation).

• Today, game theory has applications to international trade liberalization, environmental negotiations, arms control, and the shift from East/West to North/South conflict.

Next Time

2/20 Zerosum Games

Rapoport, Ch 1-7, pp: 13-93.

Nonzerosum Games

Gibbons, Ch 1, pp 1-51.

UNIT I:Overview & History Introduction: What is Game Theory? Von Neumann and the Bomb The Science of...

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