Politics and the Theory of Games Lecture

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Politics and the Theory of Games Lecture 12
The state and new institutional social sciences

• What do we want to achieve today?
• Today we introduce the state into the game.
• The state is the ultimate enforcer.
• Economists tend to assume that if the state is
  the ultimate enforcer it should enforce efficient
  property rights but of course this is not so
  simple
• The state has monopoly power and may abuse it.
• And then, of course there is the problem of
  incomplete information.
• But in the end, the structure of the state is
  incredibly complex and requires much more
  attention and careful scrutiny which is the
  subject of research for the so called
  neo-institutional social sciences, or in short
  Political Science

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Definition 1 A Game is Defined by Three
Primitives ?N,?,?

• Remember, any game in non-cooperative game theory
  is defined by three primitives
• N is the set of all relevant gents with N
  1,n and i,j ? N are generic agents in N.
• ?i ? N ?i?1,?2,,?w and ? ?1X?2XX?n
• ? ??U??n
• So we can think of many other games thus defined
  and see what they can teach us.

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Definition 2 Nash Equilibrium

• Let ?-i ?1,?2, ,?i-1,?i1,,?n ? ?
• A strategy vector ? ? ? is a Nash Equilibrium if
  and only if (iff)
• ?i ? N, ui(?) ?? ui(?-i,?i??i) ??i ? ?i

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The Social Game Without A Government

• Let Si 0,1 (with si?Si) be the set of pure
  strategies available to any agent s/he can pay
  the cost and respect the law, si1, or not, si0.
  
• Allowing for mixed strategies, a strategy space
  of agent i ? N is 0,1 with ?ipr(si1)?0,1
• So ?i is the probability that agent i respects
  the law. Let ? (?1,..., ?n) ? ? be a strategy
  vector specifying a strategy ?i that each agent i
  ? N chooses. The pay-off for every agent i ? N
  from ? ? ?, is therefore
• ui(?) ?j?i b??j - c??i?j?i ?j - ?i (since
  bc 1).

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The Unique Dominant Strategy Nash Equilibrium

• It is clear that the dominant strategy for each
  individual is not to contribute. Therefore the
  unique dominant strategy vector
• All agents maximize their utility at ?i0. So
  the unique dominant-strategy equilibrium vector
  is ?0 (0,...,0), where all agents defect with
  ui(?0)0 ? i?N.
• Note that if all agents respected the law, each
  one of them would get
• ui(?1 (1,...,1))(n-1)-1 n-2.
• If ngt2 we get an n-person version of the
  Prisoners Dilemma every agent prefers ui(?1) gt0
  over ui(?0)0, so that ?1 Pareto dominates the
  unique dominant-strategy equilibrium vector ?0.

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The question remains

• How do we get out of this mess

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Spontaneous Vrs. Political Emergence of
Institutions

• Last time we got to this point we turned right to
  what is known in the literature as the
  spontaneous approach to the emergence of social
  institutions - see in particular, Robert Sugden
  1986
• The idea here is that all institutions have to be
  explained through some initial social contract
  among equals
• Unfortunately, the Hobbesian observation that the
  only thing that emerges spontaneously is total
  chaos and bloodshed seems to be substantiated by
  all theoretical efforts to prove otherwise.

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A Game with Government Involvement

• We can present an Extensive form Game with
  Government Involvement as follows
• Suppose we let the government move first and
  decide whether or not it wants to implement (by
  force) a structure that would protect the
  property rights to its citizens. A game like
  that would look like the following
• Back ward
• induction again

Player 1- the government
Not Impose Property rights
Impose Property rights
Player 2 - the people
Respect Property rights
Not respect Property rights
1 1
0 -1
-1 -1
0 0
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Formally The Role of Government

• Introducing a government to the game we get a
  game in two stages
• Stage 1 The government chooses a strategy g
  ?0,1.
• Where g1 denotes 'enforce property rights
  and g0 denotes not to do so.
• Stage 2 Each agent chooses a probability
  ?i?0,1 with which to respect the law,
  conditional on the governments decision in stage
  1.
• The game as a triple D,G,?. A strategy for a
  constituent, di?Di is an ordered pair
  ((?i?g0),(?i?g1)) specifying the probability
• ?i?0,1 that agent i respect the law,
  conditional on whether the government enforces
  the law or not. As before we use
• D D1X...XDn and d (d1,...,dn) ? D. A
  strategy for the government g?G 0,1 specifies
  whether government enforces the law - denoted by
  g1, or not - denoted by g0.

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The Payoff Functions

• ug(d,g) t??j?N(bj ? (?i?N i?j (?i?g))) -
  (?j?g) ? cj - g ? cg
• uj(d,g)(1-t)?bj?(?i?N i?j (?i?g))-(?j?g)?cj-((
  f?g)?(1-(?j?g))

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Redefining Nash and Subgame Prefect Nash

• A strategy vector (d,g) is a NE if
• ui (d,g ) ? ui(d--i,di,g), ? i?N and ? di?Di
• ug (d,g) ? ug (d,g) ? g ? 0,1.
• A strategy vector, (d,g), is an SPNE if
• It is a NE for the entire game, as defined above
  and
• Its relevant action rules are a NE for every
  Subgame.

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The Unique Subgame Nash Equilibrium

• 1. If tn(n-1)ltcg, the unique SPNE is for the
  government not to grant the right and for agents
  to ignore each others rights.
• 2. If tn(n-1)gtcg, then the unique SPNE for the
  government to grants the right and all agents to
  respect each others rights.
• 3. If tn(n-1)cg, then both outcomes 1 and 2
  above are SPNE.

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Wednesday

• It is not as simple as it may look
• Monopoly power
• Incomplete information