Mathematical Economics II

1
Mathematical Economics II

• Housekeeping
• Bayesian Equilibrium review
• Sequential equilibrium and its refinements
• Mechanism design
• Auctions
• Bargaining and extensions

2
Housekeeping

• http//www2.warwick.ac.uk/fac/soc/economics/ug/mod
  ules/3rd/ec301/details/
• Assessment 5 seminar participation, 45
  Essay/project (problem set option for non Maths
  Econ students) 50 Examination
• Contacting me j.a.k.cave_at_warwick.ac.uk,
• (024765) 23750, Friday 1030-1130 OBA
• Need to reschedule class for week 4 (13 October),
  week 5 (20 October), week 9 (24 November) not
• Fridays 12-1
• 19-21 and 24-25 October
• 2, 21-22 and 24-5 November

3
Lectures and Seminars

• Lectures
• Weeks 1-2 Topics on incomplete information
  games (Refinements of Bayesian equilibrium
  sequential, perfect Bayesian, intuitive and
  divine equilibria - especially in signalling and
  agency games)
• Osborne ch 9, 10
• Weeks 3-5 Mechanisms and auction theory and
  practice
• Osborne sect. 9.6, online notes, Klemperer survey
• Weeks 7-8 Bargaining, Shapley value,
  applications
• Osborne ch 16, notes.
• Seminars
• Groups will lead discussions around chosen topics
  groups and topics should be arranged in class
  to facilitate team-working. You will be expected
  to read and critically discuss a number of papers
  (Ill suggest at least two for each topic).
  Participation is mandatory (5 of mark).

4
Seminar topics and Projects

• Some possible topics
• Topic(s) in signalling and/or principal agent
  games e.g. insurance, contract design,
  reputations, etc.
• Topic(s) on auctions e.g. telecom auctions
  (design, impact on market), auction-like
  mechanisms (e.g. electronic markets), etc?.
• Evolutionary methods (e.g. evolutionary games,
  games with imperfect rationality, 'herding
  behaviour', evolution of conventions)
• Network economics (e.g. transport or telecom
  networks, New Economy topics, Intellectual
  Property Rights, games of network formation or
  games played in networks)
• Game-theoretic analysis of trust
• Cooperative-game applications to cost allocation,
  antitrust, or international agreements
• Game-theoretic approaches to Intellectual
  Property Rights
• Projects
• You may do a two-term or 2 one-term projects.
  These may be based on your seminar topic (but
  group members should do distinct projects) and
  may be original projects, critical essays or
  surveys. All topics must be agreed with me.

5
Bayesian games - review(?)

• A game has incomplete information when players
  know different things about payoffs (or other
  relevant information)
• Remember information is imperfect if players
  know different things about (prior) moves.
• Applications include competition between firms
  with private information about costs and
  technology, auctions where each potential buyer
  may attach a different valuation to the item,
  negotiations with uncertainty about the other
  partys preferences or objectives, etc. in
  short, any real economic situation!
• The basic trick that lets us handle such games
  is to reduce incomplete to imperfect information
  by adding a chance player whose move chooses the
  payoffs.

6
2 strategic form examples

• Friend or foe player 2 does not know whether
  player 1 is friendly or not
• Which dilemma player 2 does not know which of
  the following Prisoners Dilemmas is being played

7
An extensive form example the Harsanyi trick
8
Intuitive definitions

• In general, such situations seem to require us to
  specify beliefs about others payoffs, beliefs
  about others beliefs about others payoff, etc.
• Convert to game of imperfect information its
  Nash equilibria are called Bayesian equilibria.
• This is because players revise their beliefs
  after being (partially) informed of Natures move
  e.g. after learning their own payoffs.

9
Reminder Bayes Rule
10
Bayes rule in action
11
Extensive form game with imperfect information

• N set of players
• H set of histories h (a1, , ak), where k may
  be infinite
• h is terminal if it is not a sub-history of any
  other (set Z)
• PH/Z ? N ( c chance player) is the player
  mapping (who plays after h, choosing an action in
  A(h). The chance players behaviour is described
  by a probability fc(?h)
• For each i, a partition Ii of hP(h) i into
  information sets Ii
• If h and h are in the same Ii, then A(h) A(h)
  A(Ii) and P(h) P(h) P(Ii).
• For each i, a preference relation over lotteries
  on Z.

12
More on extensive form games

• Equivalence of trees
• Inflation/deflation (splitting multi-move
  information sets to reflect recall)
• Adding/deleting superfluous moves
• Coalescing moves
• Interchange of moves
• Applying these rules gives identical strategic
  (matrix) forms, up to duplicate rows/columns.
  Rules ignore framing
• Mixed v. behavioural strategies
• Mixed strategies are lotteries over pure
  strategies behavioural strategies are
  independent lotteries at each information set
• In games of perfect recall, there is no loss of
  generality in limiting attention to behavioural
  strategies
• In sufficiently infinite games, the set of pure
  strategies may be too big to allow definition
  of mixed strategies.

13
How to think about Bayesian games

• Game depends on state, about which players are
  differentially and imperfectly informed
• Players have beliefs about the game, other
  players beliefs, etc. ad infinitum this
  structure of beliefs is called the players type
• Examples of independent, interdependent types
• Reformulate the game as follows
• Nature chooses players types and informs each
  player about his/her type
• Players choose strategies depending on their
  types
• In extensive form game players update beliefs
  based on observations
• Treat each type of each player as a separate
  player each type of player i plays against all
  (possible) types of other players
• Bayesian equilibrium Nash equilibrium in
  type-dependent strategies

14
Bayesian Game Example

• Two players, two types. If they are the same
  type, they play Prisoners Dilemma. If they are
  of different types, they play Battle of the
  Sexes. The joint distribution of types is given
  by ?, and the resulting game matrix is shown
  below.

15
Best replies

• Compute best replies for each type of each
  player, e.g.
• if type 1 of player 1 plays Top expected PO is
• If he plays Bottom instead ( 0), payoff is
• Simplifying and comparing, we get

• To use, apply appropriate prior probability. Ex
  independent, equally likely types, all ps ¼,
  and cut off values above are 1 for each
  strategy of pl. 1 and 2/3 for each strategy of
  player 2 each type of pl. 1 plays B unless
  opposite type of player 2 plays L with
  probability 1.
• Only pure strategy equilibria (0,0,0,0),
  (1,0,1,0), (0,1,0,1), (1,1,1,1).

16
A Cournot example

• Duopoly ?iqi(ti-qi-qj) common knowledge t11
  t2 is .75, 1.25 with probabilities p, 1-p
• 2s behaviour given by
• 1s behaviour given by
• So (unique) BE is

17
An auction example

• Two bidders in first-price auction of single
  object random assignment in case of a tie.
  Player type is private valuations i gets ti-bi
  for winning, 0 for losing
• Types ti drawn iid from uniform dist. on 0,1
• Look for symmetric BE where bids are increasing,
  C1 functions b(t). Payoff is ?i(t-b)Prbjltb.
  Because b is strictly increasing,
  PrbjltbPrbjltb.
• From j, Prbjltb Prb(tj)ltb
  Prtjltb-1(b)?(b)
• ?(b) is players valuation when bidding b (by
  uniformity, Pr?lttt). i maximises ?i
  (t-b)?(b), giving -?(b)(t-b)?(b) 0
• If b is is optimal strategy, she must be of
  type t?(b) when bidding b, so ?(b)
  ?(b)-b?(b)
• Obvious solution is ?(b) 2b, so each bids half
  his valuation.

18
Perfectness in imperfect information games

• In perfect information games, we have identified
  subgames as common knowledge independent parts of
  the game it is common knowledge that we are in a
  subgame, and no information set ever leaves a
  subgame.
• A subgame perfect equilibrium induces equilibrium
  behaviour in every sub game.
• In imperfect information games, the common
  knowledge restriction asks too much.
• However, we can still ask for strategies that
  involve rational behaviour at every information
  set.

19
Sequential rationality

• Due to presence of information sets, need to
  redefine rationality by moving from a strategy ?
  to an assessment (?, ?) consisting of a strategy
  ? and a system of beliefs ?.
• Informal definitions
• (?, ?) is sequentially rational if for every
  information set Ii, ?i(Ii) is a best reply given
  the beliefs ?.
• (?, ?) is strategically consistent if ? is
  derived from ? via Bayes Rule wherever
  applicable.
• (?, ?) is structurally consistent if ? at every
  information set is derived from some strategy ?
  via Bayes Rule.
• (?, ?) satisfies common beliefs if all players
  share the same belief about the cause of every
  unexpected event.
• The outcome of an assessment conditional on an
  information set I is a distribution O(?, ?I)
  over the set Z of terminal histories. It assumes
  independence (multiply probabilities) which is
  supported by perfect recall specifically, if h
  is in Z

20
More informal definitions

• Note uses perfect recall (if h contains h
  which contains h, then h and h lie in different
  information sets, so the event ak1 follows h
  and the event ak1 follows h are independent.
  If perfect recall fails, things get distinctly
  odd.
• Game starts at either of the probability ½ nodes.
  An assessment where ?1 ?3 End never reaches
  I2. If 2 has to move and has a belief ? that
  attaches positive probability to the histories
  (A,C1) and (B,C3), she cant compute O(?, ?I2),
  since any belief that comes from ? cannot give ?
• Informal definitions
• (?, ?) is consistent if it is the (Euclidean)
  limit of a sequence of completely mixed
  strategies and the associated beliefs (fully
  defined by Bayes Rule).
• (?, ?) is a sequential equilibrium if it is
  sequentially rational and consistent.

End
End
End
A
C1
C2
C3
1/2
I1
I2
I3
1/2
B
C3
C2
C1
End
End
End
21
Formal Definitions

• Let G (N, H, P, fC, (Ii), ()) be an
  extensive-form game of perfect recall.
• The assessment (?, ?) is sequentially rational if
  for each i and Ii in IiO(?, ?Ii) O(???, ?i,
  ?Ii) for all alternative strategies ?i of i.
• A strategy ? is completely mixed if for each
  player i, information set Ii, and move a in
  A(Ii), ?(a) gt 0.
• The assessment (?, ?) is consistent if there
  exists a sequence ((?n, ?n)) of assessments such
  that
• ((?n, ?n)) ? (?, ?) (in Euclidean space)
• for each n, ?n is completely mixed
• for each n, ?n is completely defined from ?n via
  Bayes rule.
• The assessment (?, ?) is structurally consistent
  if for each information set I there is a strategy
  ? (not necessarily the same one for every I!)
  such that
• I is reached with positive probability under ?
  and
• ?(I) is a Bayes rule belief given ?.
• The assessment (?, ?) is a sequential equilibrium
  if it is consistent and sequentially rational.

22
Examples of Sequential Equilibrium (SE)

• Equilibrium 1 ?1(L) 1 ?2(l)lt2/3 ?3(?)1
• Equilibrium 2 ?1(L) 0 ?2(l)0 ?3(?)lt1/4
• E1 is not part of any SE since 2s strategy is
  not sequentially rational (2 would be better off
  playing l, since the payoff to l conditional on
  having to move is 4)
• E2 is part of SE (?,?) if ?3(L)1/3.
  Consistency check ?1?(L)? ?2?(l)2?
  ?3?(?)?3(?)?.

• Equilibrium 1 ?1(M)1 ?2(l)0 consistency
  ?1(?,1-2?,?) ?2(?,1-?)
• Equilibrium 2 ?1(R)1 ?2(l)1
  ?2(L)gt1/2consistency ?1(?,?,1-2?) ?2(1-?,?)
• Equilibrium 3 ?1(R)1 ?2(l)gt2/5 ?2(L)1/2
  consistency ?1(?,?,1-2?)?2(?2(l)-?,1-?2(l)?
  )

23
Labour signalling example
UL
UH
?H
SH
w(e) (note ?(?L,e) ? (0,1) for e ? eH, eL
?L
SL
eL
eH

• Workers have one of two types of productivity ?H
  gt ?L. The utility of a worker of type ? who gets
  wage w and education e is w-e/?.
• A separating sequential equilibrium is a wage
  profile w(e), a belief function ?(?,e) and levels
  of education eH ? eL s.t.
• w(eH) ?H w(eL) ?L (zero profit) ?(?H, eH)
  ?(?L, eL) 1 (correct equilibrium beliefs)
  w(e) ?H ?(?H, e) ?L ?(?L, e) (sequential
  rationality for employer)
• wH-eH/?????wL-eL/????wL-eL/?????wH-eH/???(sequenti
  al rationality for workers)

24
Remarks and examples

• Every finite EGII has a sequential equilibrium
• Sequential equilibrium strategies are Nash
  equilibrium strategies
• Any sequential equilibrium of a perfect
  information game is SGPE
• Sequential equilibrium is not preserved under
  coalescenceStructural consistency
  neither implies nor is implied by
  consistencyNash equilibrium
  highlighted, but beliefs required for 3s mixed
  strategy cannot be structurally consistent

25
Further examples and Perfect Bayesian Equilibrium

• Sequentially rational, structurally consistent
  assessment that isnt sequential
  equilibriumStructurally consistent
  beliefs that make R,S,R sequentially rational 2
  assigns probability 1 to R 3 assigns probability
  1 to (L,C) 3 thinks both players deviated. Not
  consistent!
• Perfect Bayesian Equilibrium
• Consider games where every player perfectly
  observes the actions of the others, and where the
  players types are independent. In other words,
• chance picks each player is type ?i from a
  finite set ?i according to a full-support
  probability pi
• ((?i), (?i)) is the analogue of an assessment.
  ?i(?i) is type ?i of player is behavioural
  strategy. All players j share the same belief
  about player i, so beliefs ?i are probabilities
  on ?i representing what others think about is
  type. ((?i), (?i)) has
• Sequential rationality if
• Correct initial beliefs if ?i pi for each i
• Action-determined beliefs if

1
L
R
S
S
2
1,1,2
0,0,0
C
C
L
1,0,0
0,0,0
3
R
L
R
0,0,1
1,2,1
26
PBE, 2

• ((?i), (?i)) has
• Bayesian updating if
• All players j share the same (non-Bayesian)
  conjecture about player i following an unexpected
  move of i and must update it Bayesian fashion
  until I defects again. Revised conjectures only
  involve Is type.
• ((?i), (?i)) is a perfect Bayesian equilibrium
  (PBE) if it has the above properties.
• Every game with observed actions and independent
  types has a PBE each sequential equilibrium of
  such a game corresponds to a PBE but not
  conversely (PBE actions, beliefs
  highlighted)
• Can refine PBE by adding no resurrection
  condition note that sequential equilibrium,
  while generally more restrictive than PBE, may
  require resurrection

0
1
out
1
out
1
out
1
1
1
2
2
2
2
2
2
a
b
a
b
a
b
2
2
2
c
d
c
d
c
d
e
f
e
f
e
f
27
Signalling games

• An informed player (many types signaller) moves
  first, her move is observed by the uninformed
  player (one type receiver) who moves second
  then (usually) the game ends.
• Examples labour signalling, insurance,
  separating vs. pooling equilibria
• Reputations the chain-store game
• Refinement examples
• Dominance it is not reasonable to put positive
  weight on a type for whom the observed action is
  (weakly) dominated.
• Equilibrium Dominance it is not reasonable to
  put positive weight on a type for whom the
  observed action is (weakly) dominated in any
  subsequent sequential equilibrium.
• The Intuitive Criterion it is not reasonable to
  put positive weight on a type for whom the
  observed action cannot be profitable, if there is
  another type for whom the observed action could
  be profitable providing the receiver played a
  best response to the equilibrium strategy given
  beliefs that exclude the first type.
• Divine equilibrium (roughly) if one type is
  strictly more likely to send a given unexpected
  message than another type (in the sense that the
  set of responses that make this better than the
  equilibrium message for the first type is
  strictly contained in that of the second type),
  reasonable beliefs should place (much) more
  weight on the second type.

1
L
R
M
2,2
2
R
L
R
L
1,3
5,1
0,0
0,0