EITM Institutions Week

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EITM Institutions Week

• John Aldrich
• Duke University
• Arthur Lupia
• University of Michigan

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Weingast and Marshall (1988)

• M. How do legislators capture gains from trade?
• NH. Interest group and constituency pressures.
• P. Legislative institutions reflect preferences
  and transaction costs. Endogenous enforcement is
  required.
• C. The committee system makes deals credible and
  sustainable.

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Weingast and Marshall Assumptions

• Congressmen represent their districts
  politically responsive interests.
• Parties do not constrain legislators.
• Majority rule is required.
• KEY The problem of enforcement cannot be assumed
  away.

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The Committee System

• Exogenous jurisdiction.
• Monopoly agenda control.
• Requires a floor majority to succeed.
• Seniority system.
• Replacement via bidding.
• KEY Instead of trading votes, legislators trade
  committee-based property rights.

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Weingast and Marshall Findings

• The assignment mechanism succeeds in matching
  members with highly-valued jurisdictions.
• 80 of frosh get one of top 3 if no competition,
  they almost always get their request.
• Committee members are significantly more likely
  to be preference outliers.
• Committee members have greater influence, draw
  more contributions.

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Weingast and Marshall Empirical Implications

• These institutions are superior to a market
  exchange mechanism
• Negotiations can occur less frequently.
• Coalitions are more stable.
• But how do they really work?

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Equilibrium Concepts
 
 

• The equilibrium concepts build upon those of
  simpler games.
• Each subsequent concept, while more complex,
  also allows more precise conclusions from
  increasingly complex situations

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Normal Form Games

• The normal form representation of a game
  specifies
• The players in the game.
• The strategies available to each player.
• The payoff received by each player for each
  combination of strategies that could be chosen by
  the players.
• Actions are modeled as if they are chosen
  simultaneously.
• The players need not really choose
  simultaneously, it is sufficient that they act
  without knowing each others choices.

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Definition

• 1 to n players in an n-player game.
• Si player is strategy set.
• si an arbitrary element of Si.
• ui(si) player is payoff function.
• Definition The normal-form representation of an
  n-player game specifies the players strategy
  spaces S1,,Sn and their payoff functions
  u1,,un.
• We denote the game by GS1,Snu1,un.

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Ways to identify NE in order of ease.

• Identify pairs of dominant strategies.
• Eliminate dominated strategies.
• Identify stable pairs of pure strategies.
• Identify stable pairs of mixed (probabilistic)
  strategies.

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Example 1 A game with a dominated strategy.
Example 2 A more complicated game with
dominated strategies.
 
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Elimination of dominated strategies
 
Figure 1.1.1. Iterated domination produces a
solution.  
Figure 1.1.4. Iterated elimination produces no
solution.  
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Requirements for Iterated Domination

• If we want to be able to apply the process for an
  arbitrary number of steps, we need to assume that
  it is common knowledge that the players are
  rational.
• We need to assume not only that all the players
  are rational, but also that all the players know
  that all the players are rational, and that all
  the players know that all the players know that
  all the players are rational, and so on, ad
  infinitum.
• In the many cases where there is no or few
  strictly dominated strategies, the process
  produces very imprecise predictions.

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Nash Equilibrium

• In order for an equilibrium prediction to be
  correct, it is necessary that each player be
  willing to choose the strategy described in the
  equilibrium.
• Equilibrium represents the outcome of mutual and
  joint adaptation to shared circumstances.

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Definition

• In the n-player normal-form game GS1,Sn
  u1,un, the strategies (s1,sn) are a Nash
  equilibrium if, for each player i,
• si is (at least tied for) player is best
  response to the strategies specified for the n-1
  other players, (s1,si-1, si1,sn)
  ui(s1,si-1,si, si1,sn) ui(s1,si-1,
  si, si1,sn) for every feasible strategy si
  in Si
• that is, si solves max si ?Si ui(s1,si-1,
  si, si1,sn).
• If the situation is modeled accurately, NE
  represent social outcomes that are
  self-enforcing.
• Any outcome that is not a NE can be accomplished
  only by application of an external mechanism.

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NE Fun facts

• If iterated elimination of dominated strategies
  eliminates all but one strategy for each player,
  then these strategies are the unique NE.
• There can be strategies that survive iterated
  elimination of strictly dominated strategies but
  are not part of any Nash equilibrium.
• If most models are to produce a unique solution,
  the solution must be a Nash equilibrium.
• A game can have multiple Nash equilibria. The
  precision of its predictive power at such moments
  lessens.

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Figure 1.1.5. Iterated elimination produced no
solution. Find the Nash Equilibrium.  
Battle of the Sexes
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Mixed strategy NE

• A mixed strategy Nash Equilibrium does not rely
  on an player flipping coins, rolling, dice or
  otherwise choosing a strategy at random.
• Rather, we interpret player js mixed strategy as
  a statement of player is uncertainty about
  player js choice of a pure strategy.
• In games of pure conflict, where there is no pure
  strategy Nash equilibria, the mixed strategy
  equilibriums are chosen in a way to make the
  other player indifferent between all of their
  mixed strategies.
• To do otherwise is to give others the ability to
  benefit at your expense. Information provided to
  another player that makes them better off makes
  you worse off.

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Mixed Strategies

• In the normal-form game GS1,Sn u1,un,
  suppose Si si1,siK. Then a mixed strategy
  for player i is a probability distribution
  pi(pi1,pik), where 0pik 1 for k1,,K and
  pi1piK1.

Figure 1.3.2. Bottom is a best response to mixed
strategies by the column player in which 1/3
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Example 3 Divide the pie A game with two
pure-strategy Nash equilibria.
 
 
 
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Extensive Form Games

• Allows dynamic games player moves can be
  treated as sequential as well as simultaneous.
• Complete information games in which all aspects
  of the structure of the game including player
  payoff functions -- is common knowledge.
• Perfect information at each move in the game
  the player with the move knows the full history
  of the play of the game thus far.
• Imperfect information at some move the player
  with the move does not know the history of the
  game.

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Conceptual Advantage

• The central issue in all dynamic games is
  credibility.
• Backwards induction outcomes.
• Subgame perfect outcomes.
• Repeated games the main theme credible threats
  and promises about future behavior can influence
  current behavior.

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Structure of a EF Game

• The structure of a simple game of complete and
  perfect information.
• Player 1 chooses an action a1 from the feasible
  set A1.
• Player 2 observes a1 and then chooses a2 from the
  feasible set A2.
• Payoffs are u1(a1, a2) and u2(a1, a2).
• Moves occur in sequence, all previous moves are
  observed, player payoffs from each move
  combination are common knowledge.
• We solve such games by backwards induction.

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Backwards Induction

• At the second stage of the game, 2 faces the
  following problem, given the previously chosen
  action a1, maxa2?A2 u2(a1, a2).
• Assume for each a1?A1, player 2s optimization
  problem has a unique solution denoted by R2(a1).
• Since player 1 can solve player 2s problem as
  well as 2 can, player 1 should anticipate player
  2s reaction to each action a1 that 1 might take,
  so 1s problem at the first stage amounts to
  maxa1?A1 u1(a1, R2(a1)).
• (a1, R2(a1)) is the backward induction outcome
  of this game.
• Implies sophisticated rather than sincere
  behavior.
• The sequence of action can affect equilibrium
  strategies.

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Rubenstein (1982)

• Premises
• The following sequence repeats until an offer is
  accepted.
• Player 1 proposes a split.
• Player 2 accepts immediately or, after delay,
  makes a counteroffer.
• Player 1 accepts immediately or, after delay,
  makes a counteroffer.
• Players prefer money now. Discount rate ? -
  present value of a next period .

• Results
• The unique subgame perfect equilibrium is for
  Player 1 to take 100/(1?) and leave 100?/(1?)
  for Player 2, and for Player 2 to accept this
  offer and spurn any offer that is worse.
• Higher discount rates imply lower walk-away
  values in the current period.

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Rubenstein Implications

• The amount of the offer reflects the net present
  value to player 2 of playing the game.
• The less 2 likes waiting for payoffs the higher
  their discount rates the more that player 2
  will sacrifice for a payoff now.
• At d1, s1/2. No one fears the future. No one
  has an advantage.
• At d.5, s2/3.
• At d0, s1. Also true if a one-shot game where
  if player 2 rejects player 1s offer, all payoffs
  are zero.

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Requirements for BI

• The prediction depends on players knowing and
  reacting to what would happen if the game was not
  played as the equilibrium describes.
• We must assume that decision makes are interested
  in and capable of counterfactual reasoning.
• In some cases, the amount of counterfactual
  reasoning required is quite substantial.
• If people reason as if they undertake such
  calculations, then the theorys validity is not
  imperiled.
• When can we assume that people are, or act as if
  they are, capable of thinking through
  counterfactuals?

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Subgame Perfection

• A NE is subgame perfect if players strategies
  constitute a Nash Equilibrium in every subgame.
• Player 1 chooses action a1 from feasible set A1.
• Player 2 observes a1 and then chooses action a2
  from feasible set A2.
• Player 3 observes a1 and a2 and then chooses
  action a3 from feasible set A3.
• Payoffs are ui(a1,a2,a3) for i1,.,3.
• (a1, a2(a1), a3(a1, a2)) is the subgame-perfect
  outcome of this two-stage game.
• Example Legislative Bargaining game. Two
  equilibria, only one is subgame perfect.

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Backwards Induction Subgame Perfection

• The BI outcome prohibits noncredible threats
• player 1 anticipates that player 2 will respond
  optimally to any action a1 that 1 might choose,
  by playing R2(a1)
• player 1 gives no credence to threats by player 2
  to respond in ways that will not be in 2s
  self-interest when the second stage arrives.
• A NE is subgame perfect if it does not involve a
  noncredible threat.
• A dynamic game may have many NE, but the only
  subgame-perfect NE is the one associated with the
  backwards-induction outcome.

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A general result.

• Definition Given a stage game G, let G(T) denote
  the finitely repeated game in which G is played T
  times, with the outcomes of all preceding plays
  observed before the next play begins. The payoff
  for G(T) are simply the sum of the payoffs from
  the T stage games.
• Proposition If the stage game G has a unique NE
  then, for any finite T, the repeated game G(T)
  has a unique subgame perfect outcome the NE of G
  is played in every stage.

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Cooperation from Repetition?

• Proposition If GA1,Anu1,un is a static
  game of complete information with multiple NE
  then there may be subgame perfect outcomes of the
  repeated game G(T) in which, for any toutcome in stage T is not a Nash equilibrium of
  G.

The prisoners dilemma with one action added for
each player.
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• Suppose players anticipate that (Bottom, Right)
  will be the second stage outcome if the first
  stage outcome is (Cooperate, Cooperate), but that
  (Defect, Left) will be the second-stage outcome
  otherwise.
• The players, first stage interaction then amounts
  to the following one-shot game

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Implications

• Insights from one-shot games do not automatically
  transfer to repeated interactions.
• Repeated games require special assumptions about
  time.
•  
• Credible threats or promises about future
  behavior can influence current behavior.
• For some situations, subgame perfection may not
  embody a strong enough definition of credibility.

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New concepts

• In a game of incomplete information at least one
  player is uncertain about anothers payoff
  function.
• is payoff function is ui(a1,anti) where ti is
  called player is type and belongs to a set of
  possible types.
• Each type ti corresponds to a different payoff
  function that i might have.
• t-i denotes others types and p(t-iti) denote
  is belief about them given ti.

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Static Bayesian Game

• The normal-form representation of an n-player
  static Bayesian game specifies the players
  action spaces A1,An, their type spaces T1,Tn,
  their beliefs p1,pn, and their payoff functions
  u1,,un.
• Player is type, ti, is privately known by player
  i, determines player is payoff function
  u1(a1,,anti), and is a member of the set of
  possible types Ti.
• Player is belief pi(t-1ti) describes is
  uncertainty about the n-1 other players possible
  types, t-1, given is own type, ti.
• We denote this game by GA1,,AnT1,tn
  p1,,pnu1,,un.

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Strategy

• In the game GA1,,AnT1,tn p1,,pnu1,,un,
  a strategy for i is a function si(ti), where for
  each type ti ? Ti, si(ti) specifies the action
  from the feasible set Ai, that type ti would
  choose if drawn by nature.
• Separating strategy each type ti ? Ti chooses a
  different action ai ? Ai.
• Pooling strategy, all types choose the same
  action.
•  
• When deciding what to do, player i will have to
  think about what he or she would have done if
  each of the other types in Ti had been drawn.

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Standard Assumptions

• It it is common knowledge that nature draws a
  type vector t(t1,tn) according to the prior
  probability distribution p(t).
• Each players type is the result of an
  independent draw.
•  
• Players are capable of Bayesian updating.

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Bayes Theorem

• A state of the world. B event.
• Conditional probability p(BA), is the likelihood
  of B given A.
• We use Bayes Theorem to deduce the conditional
  probabilities of A given B.
• Bayes Theorem. If (Ai)i1,,n is the set of
  states of the world and B is an event, then
  p(AiB)
• Know
• The prior belief is p(A)
• The posterior belief is p(AB).

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Bayesian Nash Equilibrium

• In the static Bayesian game GA1,,AnT1,tn
  p1,,pnu1,,un, the strategies s(s1,,sn)
  are a pure strategy Bayesian-Nash equilibrium if
  for each player i and for each of is types ti ?
  Ti, si(ti) solves max ai ? Ai ?t-i?T-I
• ui(s1(t1),,si-1(ti-1),ai,si1(Ti1),sn(tn)t
  )pi(t-iti).
• That is, no player wants to change his or her
  strategy, even if the change involves only one
  action by one type.

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Gilligan and Krehbiel (1990)

• M. How is Congressional organization maintained?
• NH. Congressional organization is non-rational or
  distribution-motivated.
• P. Congress has minimal control over members and
  faces complex problems. Institutions are
  endogenous.
• C. Informational efficiency explains
  congressional organization.

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Gilligan and KrehbielKey Assumptions

• A single policy dimension.
• A distinction between policies and outcomes
  xp?.
• p is a policy, ? - uniformly distd w/mean 0.
• Risk aversion ui(x) -(x-xi)2.
• Sequence
• Legislature chooses committee members and
  transfers.
• Committee can specialize at cost k.
• Committee chooses bill.
• Legislature observes bill, chooses policy.
• Perfect Bayesian equilibrium

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Perfect Bayesian Equilibrium

• A perfect Bayesian equilibrium is a belief
  strategy pairing such that the strategies are
  sequentially rational given the beliefs and the
  beliefs are calculated from the equilibrium
  strategies by means of Bayes Theorem whenever
  possible.
• A defection from the equilibrium path does not
  increase the chance that others will play
  irrationally.
• Every finite n-person game has at least one
  perfect Bayesian equilibrium in mixed strategies.

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Gilligan and KrehbielConclusions

• As uncertainty grows, committee extremity falls.
• As expertise costs rise, so do optimal transfers.
• For extreme committees, the net benefit from
  specialization are zero.
• In this view, the parent chamber replicates
  itself in each committee to the extent possible.
• Legislative majorities defined on the single
  dimension are assumed to be the ultimate source
  of committee power.

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a rationale for restrictive arrangementsGilligan
and Krehbiel (1987)

• A legislature can choose an open rule or a closed
  rule.
• A closed rule is beneficial to the floor median
  because it allows her to control outcomes.
• It also deters persons with preferences different
  from her own from contributing to the collective
  effort (e.g., providing information).
• An open rule can be beneficial to the floor
  median when the informational gains outweigh the
  distributional losses.
• Gilligan and Krehbiel identify conditions where
  the floor media is better off relinquishing some
  of her control over outcomes.

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Snyder and Groseclose (2000)

• M. Explain party influence.
• NH. Party does not induce legislators to cast
  different votes.
• P. Parties exert influence strategically.
• 1st stage use lopsided votes to derive
  preferences.
• 2nd stage use close votes to derive influence.
• C. Party influence, particularly on procedural
  votes.

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Snyder and Groseclose Premises

• Vijajbjzicjdi?ij
• i legislators
• j roll calls
• v votes (proxy for preference/voting prob.)
• (a, b) roll call characteristic vector
• z preference parameter vector
• c party influence (D-R reward)
• d party BV

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Snyder and Groseclose Findings

• Party influence is high.
• Coefficient significant for 54 of close votes,
  9 of lopsided.
• The coefficient has the predicted sign almost
  always.
• Higher on leadership priority votes.
• Lower on moral issues, gun control.
• Not the sole product of unity or polarization.
• No significant chamber effect.

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Snyder and Groseclose (2000)

• M. Explain party influence.
• NH. Party does not induce legislators to cast
  different votes.
• What would you change?
• C. Party influence, particularly on procedural
  votes.

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Kiewiet and McCubbins (1988)

• M. Does the veto affect appropriations?
• NH. The president is most influential when he
  vetoes.
• P. (RR) One chamber (3 voters), one president,
  one play. Complete information, single-peaked
  preferences.
• C.
• T. The president can constrain Congress, but
  cannot induce greater spending.
• E. His requests have a greater impact when he
  prefers less spending.

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Kiewiet and McCubbins Empirical

• Requests and final figures for 43 agencies
  (1948-79).
• APPit?j1,2 ?jcj?j1ESTit?j2DEM?j3Ej?j4Ui?
  j5It ?it
• To improve inference pooling.
• To minimize heteroscedasticity log ratios.
• Theory supported, vetoes are position taking, the
  presidents influence is broader.

51
Epstein and OHalloran (1994)

• M. Institutional design What is the optimal
  amount of discretion for Congress?
• NH. Delegation is abdication. Ex post controls
  are necessary.
• P. Two players, one dimension, incomplete
  information, quadratic utilities, principal-agent
  framework, Bayes Nash equilibrium.
• C. Clarify the optimal amount of discretion.

52
Epstein-OHalloran Premises

• M. To acquire informational gains from agency
  expertise legislators risk distributive losses
  from bureaucratic drift.
• Premises
• Congress ideal point C0. Agents ideal point
  A0.
• Uc(X) -(X-C)2. UA(X) -(X-A)2.
• Xp ? where policy X? and ? is uniform on
  -1,1
• Congress chooses level of discretion d.
• The agent learns ?.
• The agent chooses policy p?d.
• Congress observes p and ? makes a veto choice.

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Epstein-OHalloran Conclusions

• Discretionary floor
• If 0?A?1/3?d1-A. If 1/3?A?1?d2/3.
• Veto preference
• Congress always prefers to have a veto.
• For large A, the agent prefers the same.
• The veto implies freedom.

• Uncertainty.
• Change policy range from 1,1 to -R,R
• As R grows, d grows.
• If asymmetric change.
• ? R (rel to R) ? ? d.
• If R grows more and A1/3, d shrinks.

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Epstein-OHalloran (1996)

• M. Does divided government affect policy?
• NH. Divided government does not affect the
  bureaucracies.
• P. Existing literature focuses on legislation, it
  should also attend to the bureaucracy.
• C. Divided government yields less discretion.

55
Epstein-OHalloran Premises

• Premises
• Congress ideal point C0. Presidents ideal
  point P0. Agents ideal point A.
• Ui(X) -(X-i)2.
• Xp ? where X? and ? is initially uniform on
  -1,1
• Congress chooses SQ and level of discretion d.
• President chooses A.
• The agent learns ?.
• The agent implements policy p?d.

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Epstein-OHalloran Conclusions

• Theory President AP, Congress SQ0, As P C
  diverge, d?.
• Empirical US Trade Policy 1890-1990.
• See equation p.388.
• Change variables, delegation is binary.
• Divided govt makes delegations less likely.
• Greater discretion yields lower tariffs.

57
Epstein-OHalloran (1996)

• M. Does divided government affect policy?
• NH. Divided government does not affect the
  bureaucracies.
• P. What would you change?
• C. Divided government yields less discretion.

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What are the Empirical Implications of

• Game Theory Fundamentals
• Backwards Induction
• Subgame Perfection
• Repeated Games
• Varying Equilibrium Concepts
• Adavanced Institutional Theories
• Weingast and Marshall
• Gilligan and Krehbiel
• Epstein and OHalloran