Economics 650

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Economics 650

• Game Theory

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Three-Person Games
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A Three-Nation Game
Runnistan, Soggia and Wetland each has a
shoreline Overflowing Bay. The strategies for the
three countries are the positions at which they
station their forces. Runnistan in the north or
the south Soggia in east or the west
Wetland off shore on Swampy Island, which it
controls, or on shore.
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Payoffs
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An International Alliance
In game theory, a group of players who coordinate
their strategies is called a coalition. In this
game, then, there are three possible two-player
coalitionsRunnistan and Soggia and Runnistan
and Wetland and Soggia and Wetland. We see that
any of these corresponds to a Nash equilibrium in
this case.
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Other Coalitions

• The Grand Coalition of all 3 countries.
• But that does not correspond to a Nash
  equilibrium.
• Singleton coalitions of just one member.

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Lessons

• When there are more than two players, we have to
  allow for ganging up, coalition formation.
• Coalitions can form even without enforcement.
• Such a coalition can supply a Schelling focal
  point and resolve the uncertainty in a game with
  plural equilibria.

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A "Spoiler" In A Political Game
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Stock Advising
Luvitania is a small country with an active stock
market but only one corporation, General Stuff
(GS), and only three market advisors June,
Julia, and Augusta. Whenever at least two of the
three recommend "buy" for General Stuff, the
stock goes up, and thus the advisors who
recommend "buy" gain in reputation, customers,
and payoffs.
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Stock Market Advisors
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Nash Equilibria

• There are two Nash equilibria -- the two cases
  where all three make the same recommendation, a
  self-fulfilling prophesy.
• (In the real world, stock market movements are
  not JUST self-fulfilling prophesies -- but .)

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Lessons

• Herd behavior on the part of market advisors
  may be rational.
• New information might provide a Schelling focal
  point in this game with two equilibria, leading
  to market reactions out of proportion to the
  information itself.

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A Public Goods Contribution Game 
Definition A Public Good if a good or service
has the properties that everyone in the
population enjoys the same level of service, and
it does not cost any more to provide one more
person with the same level of service, then it is
what economists call "a public good."
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The Rules

• The players are Jack, Karl and Larry (J, K, and
  L).
• Each player may choose to contribute or not
  contribute one unit of a public good. Players who
  contribute pay a cost of 1.5 units.
• If a player contributes, his payoff is the total
  number of units contributed, minus the 1.5 cost
  of the player's own contribution.
• If a player does not contribute, his payoff is
  the total number of units contributed.

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The Payoffs
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Equilibrium
Not to contribute is a dominant strategy for each
player, but this dominant strategy equilibrium is
inefficient. All three of the players would be
better off (with payoffs of 1.5 rather than 0) if
all contributed. In fact, this is another
instance of a social dilemma a three-person
social dilemma.
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New Three-Person Game Example

• This will be the game theory of dictatorship.
• In political theory, the social contract began
  with Thomas Hobbes.
• He said that the subjects of the king had a
  contract among themselves all to obey the king
  and all were better off as a result.
• The king was not a party to the contract.
• However Rousseau argued the subjects could
  get together and revise their contract. Why not?

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Obey or Resist?
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Summary on 3-Person Games, 1
Three-person games can be presented in normal
form with tables that are a bit more complicated
than those required for two-person games, but
still simple enough to get on a single page.
Three-person games bring into game theory many
issues that did not arise in two-person games,
but do arise in games with more than three
players.
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Summary on 3-Person Games, 2

• These include
• Coalitions
• Spoilers
• Crowding
• Herd behavior
• Other 2-person interactions, such as social
  dilemmas, are still observed as well

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Game Theory and Cooperation

• Game theory is interactive decision theory.
  (Schelling, Aumann).
• Game theory has two major branches
• Noncooperative
• Example Prisoners Dilemma
• Cooperative
• Coalitions can form for mutual benefit

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The Emergence of Cooperative Game Theory

• The founding book, The Theory of Games and
  Economic Behavior, originated both noncooperative
  and cooperative game theory.
• Noncooperative game theory was considered the
  solution only for two-person, zero-sum games.
• For all more complex games, their approach was
  cooperative.
• It was John Nash who extended noncooperative
  approaches to win-win and lose-lose games.

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Ideas from Cooperative Game Theory I

• Cooperative game theory draws extensively on
  mathematical set theory.
• A, B, C denotes a set comprising three elements
  -- perhaps agents in an interdependent decision
  problem.
• If A, B, C form a coalition, the coalitions
  best strategy might leave C (for example) worse
  off, in the first instance.
• In that case, C would receive a side payment to
  assure a mutual benefit.

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Example 1

• Consider a project to dam a river and supply
  water for irrigation.
• A and B are downstream and benefit.
• C is upstream, so some of Cs land is flooded by
  the dam.
• To assure a mutual benefit, C must be
  compensated, and compensated sufficiently to
  receive a share of the net benefits.
• This compensation is called a side payment.

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Ideas from Cooperative Game Theory 2

• If side payments can be made without any cost for
  the payment itself, then we can focus on the
  total value the coalition can obtain -- which can
  be distributed among the members according to
  whatever rule they choose.
• This is called the transferable utility (TU)
  assumption.

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Solutions 1

• A solution for a game in coalition function
  form should tell us
• which coalitions will form, if any, and
• how each coalition will divide its winnings among
  the members.

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Solutions 2

• Von Neumann and Morgenstern proposed a complex
  dominance criterion for solution.
• It was not considered sufficiently specific.
• A major focus of research in the 1950s and early
  1960s was to narrow the search.
• Nash, Shapley and Gillies (and, later, others)
  proposed other solution concepts.
• These all rest on various concepts of bargaining
  power and on different judgments as to when
  agents will reject an offer as a bargaining
  position.

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Cooperative Games
Definition Cooperative and noncooperative games
and solutions If the participants in a game can
make binding commitments to coordinate their
strategies then the game is cooperative, and
otherwise it is noncooperative. The solution with
coordinated strategies is a cooperative solution,
and the solution without coordination of
strategies is a noncooperative solution.
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An IT Game
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Grand Coalition
When the information system user and supplier get
together and work out a deal for an information
system, they are forming a "coalition." A
coalition consisting of all (both) the players
in the game is called the grand coalition.
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Necessary ConditionA Side Payment
Because buying and selling always means that an
enforceable agreement is made and on the basis of
the agreement, a payment changes hands. In game
theory, the payment is called a "side payment."
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Payment
Definition Transferable utility A game is said
to have transferable utility if the subjective
payoffs are closely enough correlated with money
payoffs so that transfers of money can be used to
adjust the payoffs within a coalition. Side
payments will always be possible in a game with
transferable utility but may not be possible in a
game without transferable utility.
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On That Basis

• We can rule out both "no deal" and the proven
  system as strategies. The coordinated strategies
  (advanced, advanced) yield the most total
  profits.
• The net benefits to the two participants cannot
  add up to more than 40.
• Since each participant can break even by going it
  alone, neither will accept a net less than zero.

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Visualizing
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Solutions

• Allowing for side payments, all of the points on
  the solid diagonal line are possible cooperative
  solutions.
• The range can be narrowed by
• Competitive pressures from other potential
  suppliers and users,
• Perceived fairness,
• Bargaining.

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Solution Set
Concept solution set -- von Neumann and
Morgenstern defined a complicated solution with
many possible solutions, called the solution
set. For a simple game such as this one, the set
of all efficient (Pareto optimal) coalitions and
payoffs is the solution set for the game.
Definition efficient (Pareto optimal) In
neoclassical economics, the allocation of
resources is said to be efficient, or Pareto
optimal, if no-one can be made better off without
making someone else worse off.
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Why do we study non-cooperative games at all?
Noncooperative solutions occur when participants
in the game cannot make credible commitments to
cooperative strategies. Evidently this is a very
common difficulty in many human interactions.
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Another Cooperative Game Example Taxi!
There are three taxi companies in Gotham City,
each with established customer relations
(companies that call them, kickbacks to hotel
concierges, and such) in different parts of town.
By merging, two or more of them may be able to
share costs and customer contacts and so benefit.
The three are YYellow Cab Co BwBlack-and-White
Cab Co and BBatmobiles, Inc. You are to use
methods of cooperative game theory to explore
such a merger.
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Values of Coalitions
We have the data to compute the profitability of
each merged company. The values of the potential
coalitions are
This table is called the coalition function or
characteristic function of the game.
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Coalition Function
This example also has a property many game
theorists think is correct in general it is
superadditive.
That is, if two companies merge, the value of the
merged coalition is no less than the sum of the
values of the orginal coalitions.
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Solutions

• Characteristic functions are well understood in
  mathematics and so much of the information we
  have on solutions is based on this approach.
• The coalition function approach assumes
  Transferable Utility.

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Solutions
The coalition function approach assumes
transferable utility. A candidate for solution is
a coalition and a payoff schedule.
For example, suppose the grand coalition Y,Bw,B
is formed and pays (2,2,6). This is a
candidate, but is it a solution?
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Domination 1.
A candidate fails as a solution if it is
dominated. That means that members of the
coalition can shift to another coalition and all
be better off.
For example, Y and Bw can drop out and form their
own coalition for 7, paying 3.5, 3.5. That
dominates the GC with 2,2,6.
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Domination 2.
The Grand Coalition can dominate any other
coalition because of superadditivity -- if the
payouts are right. Conditions
YBw7 YB5 BwB6 2(YBBw)18 YBBw9 4, 3, 3
will do it.
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The Core
The core of a game in coalition function form
comprises all candidate solutions that are
undominated.
In this game, the core includes the grand
coalition with any payoff schedule that
satisfies the inequalities shown before.
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Marginal Contribution 1
A solution method suggested by Shapley uses the
concept of a marginal contribution. Since the
grand coalition is efficient, we assume it will
be formed.
Suppose it is formed by adding B, Bw, and Y in
that order, and each gets what it adds to the
value of the coalition.
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Marginal Contribution 2
Thus, since B forms a singleton coalition, the
value it adds is 1. The value of B, Bw is six,
so Bw adds 5 to the value of the coalition.
Since the value of B, Bw,Y is 10, Y adds 4 to
the value of the coalition. Then the payoffs
would be 1, 5, 4 for B, Bw, Y.
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Order! Order!
But the order B, Bw, Y is arbitrary. B might
object, Why cant I come last? Then I would add
3 -- or better still, if Im second, I add 5.
Thats what Im worth!
Accordingly, the Shapley values are computed by
averaging over all possible orders in which the
players might be added.
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Shapley Value

• Accordingly, the Shapley values for B, Bw, and Y
  are 2.5, 4, 3.5. Note that
• Shapley showed that this is the only solution
  that has some nice properties, including
  symmetry.
• In this game the Shapley solution is within the
  core, but that is not always so.

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Shapley Value as Cooperative Solution

• The Shapley value has many of the properties we
  want in a solution.
• When we have spoken of cooperative solutions
  before, as in the Social Dilemmas, the Shapley
  Value fits as the cooperative solution we mean.
• However, it does not always agree with the
  Core.
• Moreover, there are other cooperative solution
  concepts that may disagree with either.
• This is a problem for cooperative game theory!

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A Stag Hunt and a Problem
Lets return to the Stag Hunt Game. Without
looking at strategies, we assume that there are 3
hunters, A, B, C, and any 2 can catch a stag. 3
can catch both a stag and a rabbit.
A stag is worth five and a rabbit is worth 1.
Therefore, any 2 person coalition is worth 5, the
grand coalition with 6, while a singleton is
worth only 1.
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Stag Hunt -- Core 1
This Coalition Function is superadditive. Suppose
the grand coalition forms, with payoffs 2,2,2.
Then any 2-person coalition dominates the grand
coalition, since each 2-person coalition can
improve its total payoff by expelling the third
and hunting as a pair.
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Stag Hunt -- Core 2
Conditions for the stability of the grand
coalition.
AB5 BC5 AC5 2(ABC)15 ABC7.5 is
necessary for the GC to guard against 2-person
secessions. But this is not possible.
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Stag Hunt -- Core 2
Now suppose coalition AB forms, with payoffs 2.5,
2.5. C, who is left out, approaches A with the
following proposition
Lets form coalition AC and I will let you
take 3 -- Ill only take 2. Both would be better
off, so this dominates. To be stable AB must
pay a total of AC-C BC-C44 --
impossible. Similarly, all 2-person coalitions
are dominated.
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Empty Core

• Singletons are also dominated by all 2 or 3
  person coalitions.
• There are no undominated coalitions.
• The core in this case is the null set -- it is an
  empty core.
• This is a recognized limitation of the core
  concept.
• Note, by the way, that if a stag is taken this
  game really has no individual payoffs until the
  payoff schedule is cooperatively determined. The
  noncooperative solution is not really determined
  unless the cooperative solution is. Problem!

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Stag Hunt -- Shapley
However, we can calculate a Shapley value for
this game, as we can for any superadditive game
in coalition function form.
Since the players are symmetrical, the Shapley
value calls for equal division -- 2 each. Note
that this is not in the core -- the core is
empty.
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Problems 2

• If the TU assumption doesnt apply, we have
  another sort of problem.
• Anna, Bob, Carole and Don are all employed at the
  University of West Philadelphia (UWP) and commute
  by car from their homes in the western suburbs of
  Philadelphia to UWP.
• They are interested in forming one or more
  carpools to commute together.
• We will treat the carpools as coalitions in a
  cooperative game.
• Payoffs are in miles adjusted for gas saving --
  the objective is to minimize.

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The Coalition Function
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Dominance
Suppose ACD have a carpool and propose to add
Bob. Adding Bob will make Anna and Carole better
off (lower overall time and gas costs) and leave
Don no worse off. Thus, the Grand Coalition
weakly dominates ACD.
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Core
Reasoning in this way we find that the core of
the carpool game consists of coalitions ABD and
BCD -- meaning either Anna or Don is out of luck!
In fact, the grand coalition, despite its
overall advantages, is not in the core. It is
dominated both by ABD, which makes A and D better
off while B is no worse off, and by line 5, which
makes BCD all better off.
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Shapley?
The Shapley value, as we have defined it, only
works if utility is transferable. There are
proposals for extension of the Shapley value to
NTU games, but they are little used and will be
beyond our scope.
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Homework for Review

• Ch. 7 3,4
• Ch. 16 1,4 -- but 1 needs revision
• New part c Would a division of the catch in the
  proportions 4,1,1 (in the order GMP) be a stable
  side-payment schedule? Why or why not? Would a
  division of the catch in the proportions 2,2,2

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Summary 1.

• When players can commit themselves credibly to
  coordinate their strategies, they can often
  improve their payoffs.
• There are a number of solution concepts for this
  case.

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Summary 2.

• Some payoffs may be ruled out if we also
  consider stability against defection by smaller
  coalitions, focusing on "the core."
• However, this may rule out all of them or
  leave more than one to choose among, after all.
• Nevertheless, the core seems to describe some
  key economic phenomena.

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Summary 3.

• The Shapley value is the individuals marginal
  contribution, averaged over all orders in which
  the agents might be added to the grand coalition.
• When we speak of a cooperative solution to a
  symmetrical noncooperative game like a social
  dilemma, the Shapley value seems to be what we
  usually mean.