Microeconomics 3

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Microeconomics 3

• Mushtaq H. Khan
• Department of Economics

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Game Theory
Game Theory analyses how rational decisions are
made when decisions are interactive This means
that each individuals best choice depends on the
choices made by other individuals.
Thus in the cases discussed by game theory, each
individuals perception or belief about what
others might do has a direct impact on that
persons choice. Beliefs and perceptions now
matter, in a way they do not in conventional
economic theory.
In the conventional model of the market economy,
each individuals best choice depends only on
that individuals preferences and costs, it does
not depend on the choices made by others except
to the extent that those choices determine market
prices.
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Game theory versus Conventional Economics

• Game theory tells us that rational individuals
  know that other individuals are also rational and
  will do what is in their best interest so each
  individual will choose an action that maximizes
  their benefit given that other individuals will
  be doing the same
• Paradoxically this gives very different results
  from market economic theory where individual
  maximization always results in social
  maximization
• In game theory social maximization does not
  always follow from rational individual
  maximization. In market economics, individual
  maximization should lead to social maximization
  as long as transaction costs are low

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The Elements of Game Theory
1. The players. These are the individuals, i, who
make decisions. Each player's goal is to maximize
his/her utility by a choice of actions.
2. The actions of each player ai are the choices
they can make.
3. The information set for each player is the
knowledge he/she has at each stage of the game of
the values of different variables.
4. A player's strategy si are the actions player
i chooses at each stage of the game, given
his/her information set.
5. A player's payoff ?i(s1,s2,..sn) is the
expected utility of the player given the
strategies chosen by him/herself and all other
players.
6. The equilibrium s(s1,...,sn) is a strategy
combination consisting of a best strategy for
each of the n players in a game. There are a
number of different equilibrium concepts in game
theory which we will discuss in turn.
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The core components of Game theory

• A game is a set of rules known to all players
  according to which the players can get different
  payoffs by selecting different strategies.
• To make the modelling tractable, the rules
  themselves are fixed and cannot be changed by the
  players. All rules and payoffs are known by all
  players
• However, the payoff for each player can depend on
  the choices of other players
• This makes game theory different from the theory
  of individual choice in conventional economics
  where the choice made by an individual does not
  depend on choices made by other individuals
  (except in so far as the latter determine the
  prices faced by the individual when making
  his/her choice)
• Game theory analyses how rational individuals
  will make their choices in such an interactive
  context

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Cooperative and Non-Cooperative Games
Cooperative games are those where players can
make binding agreements with each other about the
strategies they will play. In contrast,
non-cooperative game theory assumes that
individuals will not operate in any way which is
not individually rational.
Thus cooperative games assume effective
external enforcement of contracts.
Non-cooperative games assume contracts cannot
be enforced unless the individual wants the
contract to be enforced.
Game theory is for the most part about
non-cooperative games. Cooperative games should
not be confused with cooperation in
non-cooperative games where individuals can
cooperate with each other because that is the
best strategy for them in that game.
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Equilibrium Concepts in Game Theory
A number of equilibrium concepts are used in game
theory. The best known are the Dominant Strategy
and Nash Equilibrium concepts.
In discussing equilibria we use the notation s-i
to denote the vector of strategies of all players
other than i.
Player i's best response to the strategies s-i
chosen by other players is the strategy s that
yields the greatest payoff, p i (si,s-i) p
i (si', s-i) si' ? si
The strategy si is a dominant strategy if it is
a player's strictly best response to any
strategies the other players may pick, in the
sense that whatever strategies they pick, his
payoff is highest with si, p i (si,s-i) p i
(si',s-i) for s-i, and si' ? si
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The Nash Equilibrium Concept
The strategy combination s is a Nash Equilibrium
if no player has an incentive to deviate from his
strategy given that the other players do not
deviate, i, pi(si,s-i) pi(si',s-i)
si'.
Note that the definition of Nash equilibrium
lacks the extra s-i of the dominant strategy
equilibrium so a Nash strategy only needs to be
the best response to other Nash strategies, not
to all possible strategies. This is also shown in
the definition by the fact that the Nash
maximizes pi(si,s-i) and not pi(si,s-i).
Thus the Nash is a best response on the
assumption that other players are playing Nash.
Hence Varian argues that the Nash equilibrium is
an equilibrium in actions and beliefs.
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The Nash equilibrium

• In the Nash equilibrium each person asks
• If I follow a particular strategy, assuming that
  others are following a particular strategy, will
  they continue to follow their particular
  strategies if they know that I am following this
  strategy?
• If the answer to this question is yes for every
  individual following a strategy, the combination
  of strategies for all individuals is a Nash
  equilibrium for the game
• The characteristic of a Nash equilibrium is that
  once everyone is playing a Nash, no-one will play
  any other strategy, even if the Nash does not
  appear rational for all players collectively
• There may be more than 1 Nash equilibrium for a
  game, and the outcome may be socially undesirable
  because with more than one equilibrium, rational
  players can make mistakes and end up with
  suboptimal outcomes

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Justifications for the Nash Equilibrium Concept
1. It gives a formal definition of equilibrium if
there is a unique self-evident way to play a
game. If there is an equilibrium to a game, it
must be because rational players have no
incentive to move away from it.
2. The Nash may represent a self-enforcing
convention. If the players can engage in
(non-binding) communication before the game then
they will have to work out a strategy combination
which is Nash.
However, the Nash has a number of problems
1. There may be more than one possible Nash
equilibrium. Game theory does not tell us how
equilibria are selected.
2. The Nash equilibrium solution may be so
complex that it is difficult to argue that the
solution is the self-evident equilibrium which
would be chosen by rational actors.
3. The Nash simply describes the likely
conventions which emerge. It does not explain the
emergence of beliefs about what other agents will
do. It is thus descriptive rather than analytical.
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The Dominant Strategy equilibrium

• Some games (like the prisoners dilemma) have a
  stronger equilibrium called the Dominant Strategy
  equilibrium
• Here, each player asks if there is a strategy for
  that player that is the best strategy regardless
  of the particular strategies played by the other
  players.
• If it exists, such a strategy is called a
  dominant strategy for that player
• If all players have a dominant strategy, the game
  has a dominant strategy equilibrium, and these
  equilibria are very stable
• All dominant strategy equilibria are also Nash
  equilibria, but not vice versa

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The Payoff Structure in a Simple Game

• In a 2 person game we write the Row players
  payoff first, the Column players payoff second

B
Drive on Left
Drive on Right
Drive on Left
1,1
-10,-10
A
1,1
-10,-10
Drive on Right
How will rational individuals decide what to do?
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Nash Equilibria in a Coordination Game
B
Drive on Right
Drive on Left
Drive on Left
1,
1
,-10
-10
Not NE
Nash Equilibrium
A
-10,
1
1,
-10
Drive on Right
Not NE
Nash Equilibrium
Thus in this simple coordination game there are 2
Nash equilibria but rational individuals can make
mistakes ending up with -10,-10 There are no
dominant strategies in this game
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Extensive Form of (Coordination Game)
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Identifying Nash Equilibrium
Take each strategy combination like (Left,
Right), and ask i) will A play Left if B plays
Right and ii) will B play Right if A plays Left.
If the answers to both questions are YES, (Left,
Right) is a Nash equilibrium Do this for all
possible strategy combinations
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The Prisoners Dilemma Game
The most widely studied problem involved in game
theory is that of free-riding. This is the well
known Prisoner's Dilemma model. Free-riding is a
structure of payoffs where there are gains from
cooperation but where the individual gains are
even greater if everyone cooperates but the
individual cheats.
With Prisoner's Dilemma payoffs the private
incentive structure creates no incentive for
cooperative behaviour and mutually gainful
cooperation now requires stronger conditions. The
Nash equilibrium here is not the pareto efficient
one.
The Prisoner's Dilemma problem emerges when the
payoff structure is such that i) cooperation is
the socially optimal strategy, so unlike the
Chicken Game, there is a unique outcome which is
socially desirable, ii) the best outcome for each
individual is that the other person cooperates
and he defects, iii) the worst outcome is that he
cooperates and the other defects.
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The Payoff Structure in a Prisoners dilemma
B
Cooperate
Defect
Cooperate
0,10
8,8
Lowest payoff for A but highest payoff for B
Highest joint payoff
A
1,1
10,0
Defect
Highest payoff for A but lowest payoff for B
Lowest joint payoff
What matters is not the actual payoffs but their
ranking for each player
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Nash Equilibrium in the Prisoners dilemma
B
Defect
Cooperate
Cooperate
8,
8
,10
0
Not Nash Equilibrium
Not NE
A
10,
1
0
1,
Defect
Not NE
Nash Equilibrium
Defect, Defect is not only a Nash equilibrium, it
is a Dominant Strategy Equilibrium as well and is
therefore very stable
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Prisoners Dilemma 2
Non-cooperation now becomes the Dominant
Strategy, ie whatever anyone else does, the best
strategy for each individual is to defect, or
choose the socially sub-optimal option.
Examples of prisoners dilemma games i) The
Creation of Social Order Hobbes and the Theory
of the Leviathan
ii) Cartels Adam Smith and the Free-Market
Doctrine
iii) The Provision of Public Goods
iv) Worker-Capitalist Cooperation Leibenstein
and the Explanation of Differences in Capitalist
Performance based on Trust
v) As a component of all types of collective
action problem
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Escapes from the dominant strategy equilibrium

• Rational individuals will harm themselves by
  being forced into the dominant strategy
  equilibrium by their fear of being ripped off by
  other rational individuals
• There are three escapes from the prisoners
  dilemma
• i) External enforcement. This is where the role
  of the state comes in as external enforcer. Weber
  defined the state as the body with a legitimate
  monopoly of violence
• ii) Repeated games. If the same players play the
  game for an indefinite number of times,
  cooperation may emerge as a voluntary strategy if
  some players can credibly threaten to punish
  those who defect by not playing cooperate
  strategies with them
• iii) Less than rational players, who play on the
  basis of trust or lack of full information about
  the rationality of other players