Economics 650

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Economics 650

• Game Theory

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Sequential Games
While all games can be treated in "normal form,"
there are other important interactions in which
the agents have to choose their strategies in
some particular order, and in which commitments
can only be made under limited circumstances or
after some time has passed. These are "sequential
games."
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Strategic Investment to Deter Entry
The entry of new competition can reduce the
profits of established firms. Accordingly, we
would expect that companies might try to find
some way to prevent or deter the entry of new
competition into the market, even if it is costly
to do so. Here is an example of that kind.
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The Chips are Down
Spizella Corp. produces specialized computer
processing chips for workstations. A plant to
fabricate these chips costs 1 billion and will
produce 3 million chips per year at a cost of 1
billion per year and so at an average cost of
333.33 per chip.
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Demand for Chips
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A Challenge
Spizella's management have learned that Passer,
Ltd, are considering building a "fab" to enter
this market in competition with Spizella.But if a
second fab comes on line, output will increase to
6 million chips, the price per chip will drop to
400, reducing Spizellas profits.
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Threat to Profits
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Response
Nevertheless, Spizella is considering investing
in a second fab.
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Why?
1) If Spizella builds before Passer makes their
decision, Passer will realize that their plant
would be the third one, and that if they build it
everyone will lose 400 million per plant per
year. So Passer will not build, and Spizella will
retain 400 million a year of profit. 2) If
Spizella doesn't build the second plant, Passer
will, and Spizella will be left with only 200
million of profits on their one present fab.
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Strategic Investment
In building the new plant to keep Passer out,
Spizella would be engaging in "strategic
investment to deter entry."
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In Extensive Form
A Subgame
Another Subgame
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Subgames
We see two subgames of this game of strategic
investment to deter entry. In game theory, the
whole game is also considered a subgame -- of
itself. The others are proper subgames.
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Perfect Equilibrium
In order for a game with one or more proper
subgames to be in equilibrium, every subgame must
be in equilibrium. This sort of equilibrium we
call a "subgame perfect equilibrium."
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Solution by Backward Induction
We start with the last decision in each sequence,
determine the equilibrium for that decision, and
then move back, determining the equilibrium at
each step, until we arrive at the first decision.
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Reducing
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Recapitulating Concepts --
Definition Subgame --A subgame of any game
consists of all nodes and payoffs that follow a
complete information node.
Definition Proper Subgame --A proper subgame is
a subgame that includes only part of the complete
game.
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The Prisoners Dilemma Has No Proper Subgames
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More Concepts
Definition Basic and Complex Nodes -- A node is
"basic" if each of its branches leads to just one
set of payoffs. (This means, in effect that there
are no further decisions to be made.) A node is
complex if it is not basic.
Definition Subgame Perfect Equilibrium -- A game
is in subgame perfect equilibrium if and only if
every subgame is in a Nash equilibrium.
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Backward Induction
Method Backward induction To find the subgame
perfect equilibrium of a sequential game, first
determine the Nash Equilibria of all basic
subgames. Next, reduce the game by substituting
the equilibrium payoffs for the basic subgames.
Repeat this procedure until there are no proper
subgames, and solve the resulting game for its
Nash Equilibrium. The sequence of Nash Equilibria
for the proper subgames of the original game
constitutes the subgame perfect equilibrium of
the whole game.
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The Spanish Rebellion
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Reduced
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The Centipede
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Reduced Centipede
The game has just one proper subgame, shown by
the gray oval, and it is basic. In that subgame,
B will choose "grab," and the payoff will be 2
for A and 6 for Barb. With this solution we
reduce the Centipede game to the game shown here.
In this game Anna chooses "grab" for 4 rather
than "pass" for 2, and that is the subgame
perfect solution to the game.
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Lessons

• Backward reduction can give us a solution to a
  game in extensive form.
• In some cases, people may want to give up their
  freedom of action by making an early commitment.
• Early commitment may improve payoffs -- or not.

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A Defense Application
During the cold war period the Soviet Union had
superior numbers of ground troops in Europe and
it was generally believed that if they chose to
attack, they would have been able to overrun West
Germany very rapidly. To prevent this, U.S.
troops were stationed on German soil. But the
American troops were not strong enough to defeat
an all-out Soviet attack. Thus, the U. S.
stationed in Germany a force that the Soviets
could surely defeat if they wished. Why?
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Answer?
People at the time said that the troops were a
tripwire.
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The Cold War on the European Frontier Without
Troops in Germany
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The Cold War on the European Frontier With Troops
in Germany
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Nested

• Part of the idea was that the American electorate
  would not permit a government that would fail to
  retaliate if American troops were attacked.
• Thus the US changed the rules by nesting the game
  against the SU in a larger game.

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Nested and Imbedded

• Nested games If a game is part of a larger
  game, then equilibrium strategies in the smaller
  game may depend on the larger game. The smaller
  game is said to be nested within the larger.
• Imbedded games If a nested game is a proper
  subgame of the larger game, then the nested game
  must be in equilibrium for the larger game to be
  in a subgame perfect equilibrium. Then the
  smaller game is said to be imbedded in the
  larger.

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Changing the Rules
In effect, the U. S. changed the rules by
imbedding the original game in a larger game.
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Planning Doctoral Study
Nora can infer that Anna plans to study SE, since
otherwise Anna would keep her outside option,
that is, her job. This is sometimes called
forward induction.
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The Basic Proper Subgame
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A New Example 1

• This game begins with a decision of a municipal
  authority, M, either to build a public parking
  lot (Build) or not to build it (Dont).
• In either case, at the second stage two
  businesses, B1 and B2, decide whether or open
  retail sites near the potential parking lot
  (Build) or not to do so (Dont).
• They have to decide simultaneously
• The businesses are complementary rather than
  competitive, as, for example, one offers gourmet
  groceries and the other wine and spirits.

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A New Example 2

• If the municipal public parking lot has been
  built, then either or both can operate profitably
  (with a payoff indicated as 4)
• if the public lot has not been built, each will
  have to supply parking space for their own
  customers, raising their costs
• Nevertheless, if both build, they can both
  operate profitably, with payoffs at 2, 2.
• If only one builds, however, it will not attract
  enough customers to cover the cost of the parking
  lot and will operate at a loss indicated by 1.

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A New Example 3

• A business that does not build has a payoff of
  zero.
• If both businesses build with their own parking
  lots, the municipality has the further
  possibility of imposing a regulation that a
  proportion of all parking places must be public
  parking, and supposing that this public parking
  will attract more customers away from the rival
  Mall in a nearby township, the profits of both
  businesses are higher, at 3.
• Since the municipality has no sources of revenue
  (other than taxes) its payoffs are always
  nonpositive, and the cost of the public parking
  lot is 2 while the cost of administering a
  parking regulation is 1.

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The payoffs are in the order B1, B2, M.
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Proper subgames
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A Nested (not Imbedded) Game
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An Imbedded Game
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The Imbedded Game Reduced
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The Imbedded Game in Normal Form
We observe that this subgame has 2 Nash
equilibria. So has the other imbedded game
between the two businesses.
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Concluding the Example

• Since payoffs for the municipality are always
  nonpositive, its dominant strategy is never to do
  anything.
• Thus there are two subgame perfect equilibria,
  corresponding to the two solutions we have just
  seen.
• For a cooperative solution, we will require a
  side payment (i.e. taxes) to the municipality.
• Note, though, that it is useful to think of
  private sector decisions as games imbedded in the
  larger game of public policy.

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Just Fueling Around
Transport Equipment Corp. (TECORP) sells busses
primarily to urban bus services and wants to
convert its busses to fuel cell power since its
customers are concerned about air pollution.
Queen Hill Power (QHP) has perfected the
technology for the fuel cells and TECORP has
approached QHP to produce power plants for its
busses.
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Queen Hills Problem
However, this will require QHP to construct a
costly, specialized facility for which TECORP
will be the only buyer. QHP is concerned that
TECORP will then demand a renegotiated price,
which QHP will be forced to grant, and thus be a
loser.
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Payoffs

• if there is no agreement, payoffs are 0,
• if there is an agreement and
• no regenotiation, payoffs are 100, 100
• a demand for renegotiation and QHP refuses,
  payoffs are 0, -100
• a demand for renegotiation and QHP gives in,
  payoffs are 200, -50 (TECORP first)

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Subgame Perfect
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How about a merger?
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Why Nonprofits?1

• Nonprofit enterprise is a faster-growing sector
  of our economy than investor-owned, cooperative
  or government enterprise.
• Nonprofits may actually earn profits, but may not
  distribute the profits.
• Instead, any profits must be devoted to the
  nonprofit corporations mission.
• Why should such enterprises exist?
• Many nonprofits are supported or founded by
  donations.

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Why Nonprofits? 2
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Partly Reduced
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Partly Reduced
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Further Reduced
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Overall Summary

• Use the concept of subgame perfect equilibrium.
• We use backward induction.
• Think forward and reason backward.
• We first solve all of the basic proper subgames
  -- continue step by step.
• Sequential commitment makes a difference!