Static Games of Incomplete Information

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Static Games of Incomplete Information

• .

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Example 1

• Two firms incumbent (player 1) entrant (player
  2)
• Player 1 decides whether to build new plant
  player 2 decides whether to enter
• Player 1s cost of building is 1.5 (wp 1-p1) or 3
  (wp p1)
• How is this game to be played?
• It depends on 2s belief about 1s probability of
  building when cost is low (x), and 1s belief of
  2s probability of entering (y)

Enter Dont enter
Build 0, -1 2, 0
Dont build 2, 1 3, 0
Enter Dont enter
Build 1.5, -1 3.5, 0
Dont build 2, 1 3, 0
2
2
1
1
1s cost is high
1s cost is low
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Example 1

• Harsanyis Key Idea (1967-68)
• i. Player 1s type is determined by a prior
  move of nature
• ii. Transforms a game of incomplete info (2
  doesnt know 1s cost) into one of imperfect info
• Equilibrium 1. Player 2 enters (y1) if
  xlt1/2(1-p1, and stays out (y0) if
  xgt1/2(1-p1,
• 2. Low cost player 1 builds (x1) if
  ylt1/2, and not (x0) if ygt1/2.

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Bayesian equilibrium

• There are i ?I players
• Players types are drawn from dist p(?1, ?2,
  ?I,), where ?i ?Ti, Ti is finite and p(?-i ?i)
  is is conditional probability about his rivals
  types ?-i
• Pure strategies for i are si ?Si, and payoff is
  ui(s1, s2, ,sI, ?I, ?2,,?I,),
• A Bayesian equilibrium for above game of
  incomplete information is a Nash equil of the
  expanded game where each player is space of
  pure strategies is the set of maps from Ti
  to Si. Given strategy profile s(.), and s/i(.) ?
  , s(.) is a Bayesian equilibrium if

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Cournot competition with incomplete info

• Let firm is profit be ui qi(?i- qi- qj), where
  ?ia-ci, a being the intercept of the linear
  demand function
• Let ?11, and ?23/4 w.p. ½ and 5/4 w.p. ½
• Denote q2 as q2H if ?23/4 , and as q2L if
  ?25/4
• Firm 2s equil choice must satisfy q2(?2)(?2-
  q1)/2
• Firm 1 does not know 2s type, so its expected
  payoff is
• This gives q1(2- q2L - q2H )/4
• Plugging in for q2(?2) we get (q11/3, q2L
  11/24, q2H 5/24)
• This is the Bayesian equilibrium

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War of attrition

• Each player i chooses a number si in 0, 8,
• Payoffs are
• is type ?i takes values in 0, 8, with
  distribution function P and density p
• We look for pure-strategy Bayesian equil (s1(.),
  s2(.))
• For each ?i, si(?i ) must satisfy

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War of attrition

• A. Key step Look for monotonic strategies that
  are strictly increasing and continuous in a
  players type
• 1. To show that If ?i// gt ?i/ implies si //gt si
  / where
• si //si (?i //) and si /si (?i /)
• Proof If ?i/ prefers si (?i /) to si (?i //)
  then
• A similar inequality obtains when ?i// prefers
  si (?i //) to si (?i /)
• Subtracting the two inequalities gives
• Since, ?i// gt ?i/ it must be that si //gt si / .
  

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War of attrition

• 2. To show that strategies, si (?i) and sj (?j),
  are strictly increasing
• Proof If not, there would be an atom for j at
  sgt0, i.e. Pr(sj (?j)s)gt0
• Then i assigns probability 0 to s-?, s
• Then any type of j planning to play at s is
  better-off playing at s-?
• Thus, no atom at s after all
• 3. To show that si (?i) is continuous in ?i
• Proof Similar to above
• B. Let Fi be inverse function of si F-1i (?i)
  si

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War of Attrition

• C. Transforming variable of integration from ?j
  to sj,
• D. The FOC If si si (?i), then i cannot
  benefit by playing si dsi instead of si
• Cost is dsi if j plays above si dsi which has
  probability
• 1-Pj(Fj(si dsi ))
• Expected cost, to first order in dsi , is
  1-Pj(Fj(si )) dsi .
• Gain is ?i Fi(si) if j plays in si, si dsi ,
  i.e. if
• ?i ? Fj(si ), Fj(si dsi )
• This has probability, pj(Fj(si ))F/j(si ) dsi .
• Equating costs and benefits, the FOC is
• Fi(si) pj(Fj(si ))F/j(si ) 1-Pj(Fj(si ))

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War of Attrition

• Impose symmetry P1 P2P
• Substitute ? F(s), and use F/1/s/ to get,
• Integrating,
• If P(.) is exponential, P(?)1-exp(-?), then,
• S(?) ?2/2

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Double auction

• A seller and buyer trade a unit of good
• Seller (player 1) has cost c and buyer (player 2)
  has valuation v . v, c ?0, 1
• Players simultaneously bid b1, b2 ?0, 1
• If b1 b2 , they trade at price t (b1b2)/2
• With trade 1 gets u1(b1b2)/2-c 2 gets
  u2v-(b1b2)/2. Without trade both get 0
• c distributed as P1 and v distributed as P2
• Let F1(.) and F2(.) be cumulative dist of b1 and
  b2
• Find the Bayes-Nash equilibrium (s1(.), s2(.)),
  where si(.)0, 1?0, 1

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Double auction

• To show that Bids increase in type. That is, c//
  gt c/ implies b1//gtb1/ where b1 //s1(c//) and
  b1/s1(c /)
• Proof Optimization by the seller requires
• and
• Combining these inequalities,
• Since, c// gt c/ it must be that b1//gtb1/ .
• The bids are strictly increasing and continuous
  in types
• Similar things hold for the buyer

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Double auction

• Maximization problem for type-c seller
• The FOC is ½ 1-F2(s1(c))-(s1(c)-c)f2(s1(c))0
• For the buyer,
• And FOC is (v s2(v)f1(s2(v)) ½ F1(s2(v))

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Double auction

• Specific case
• - P1 and P2 are uniform dist on 0, 1, and
  strategies are linear in types s1(c) a1ß1c
  and s2(v) a2ß2v
• - Then, Fi(b)Pi(s-1i(b)) s-1i(b)(b- ai)/ßi,
  so fi(b)1/ ßi,
• - Plugging into FOCs, we get
• 2a1(ß1-1)c/ ß2 ß2 (a1ß1c) a2/ß2
• 2(1-ß2)v a2/ ß1 a2ß2v- a1/ß1
• - Solving this system, ß1 ß2 2/3 a11/4
  a21/12
• - In equilibrium parties trade only if, a2ß2v
  a1ß1c
• - Thus trade occurs only if, v c1/4
• Too little trading in equilibrium!!

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First price auction with a continuum of types

• Two bidders with a unit of good to trade
• Player is valuation is ?i and belongs to
• Players have beliefs P, with density p, about
  rivals valuation
• Seller imposes reservation price s0gt
• Player i bids si gets ui ?i - si if si gt sj
  ui 0, si lt sj
• If si sj both get good w.p. ½ and ui (?i -
  si )/2
• Let si (.) be the pure strategy of player i

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First price auction (continuum of types)

• Show that strategies are monotonic, strictly
  increasing, and continuous in type
• To show that
• Proof If then type of player i could
  slightly lower his bid and still win w.p. 1
• Let Fi be inverse function of si (.) F-1i (?i)
  s on
• s0, , i.e. player i bids s if his
  valuation is Fi(s)
• Type ?i maximizes (?i - s)P(Fj(s)) over s
• This gives, P(Fj(s)) Fi(s)-sp(Fj(s)) Fj/(s)
• There is a similar FOC by switching i and j

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First price auction (continuum of types)

• To show that There cannot be an asymmetric
  solution, F1(s) ? F2(s) for all s
• Using F1 F2 F, in FOC, and integrating, we
  get,
• This will give F(.), and the inverse function
  gives s(.)

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First price auction (with two types)

• Each bidder can have types , with lt
  
• Corresponding probabilities are and
• Sellers reservation bid is lower than
• Key idea Look for mixed-strategy equilibrium
• Type bids , and type randomizes
  according to a continuous distribution F(s) on
• Argue that
• For i of type to play a mixed strategy with
  support
• it must be that

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First price auction (with two types)

• Because F( )0, the constant is
• Thus, F(.) is given by
• Let G(s) be distribution of bids. Above
  can be written as
• Since F( )1, implies
• Each bidders net utility is 0 when his type is
  and
• when his type is

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Bayesian equil can justify mixed equil

• Harsanyi, 1973 A mixed strategy equil of a
  complete info game can be interpreted as the
  limit of pure strategy equil of perturbed games
  of incomplete info
• Example Grab the Dollar- complete info version
• - At times t0, 1, 2, two players want to grab
  a 1
• - If only one grabs, he gets 1 and other gets 0
• - If both grab at once, dollar destroyed each
  gets -1
• - If neither grabs, both get 0
• - Players have a common discount factor d
• - The only symmetric strategy is a mixed
  strategy, where both grab w.p. p1/2 in each
  period

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Bayesian equil can justify mixed equil

• Example Grab the dollar- complete info version
• -Consider player 1. By grabbing at time t, he
  gets
• dt(1- p) dt p(-1). By not grabbing, he gets
  0. He is indifferent, so dt(1- p) dt p(-1)0,
  and so, p1/2
• Consider perturbed version of the above game
• Example Grab the dollar- incomplete info
  version
• -If player i wins, he gets 1?i, ?i is uniform
  on -?, ?
• -Consider symmetric strategy si(?ilt0)do not
  grab si(?i0)grab
• -This is a pure strategy Bayesian equilibrium!
  Why?
• -What happens when ??0?

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Bayesian mixed equil 1st price auctions

• 1. Consider FOC for first-price auctions with
  continuum of types P(Fj(s)) Fi(s)-sp(Fj(s))Fj
  /(s)
• Let Gj(s) P(Fj(s)) be dist of bids s.
  gj(s)p(Fj(s))Fj/(s)
• Then FOC becomes Gj(s) Fi(s)-s gj(s), sgt
• 2. Equivalent condition for two-type case
• Differentiating w.r.t s,
• Consider sequence, Pn(?), s.t.
• If Fn(.) is equil strategy for Pn(.), then