Job Market Signaling
(Spence model)
 Perfectly competitive firms are bidding for services of
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workers. Competition bids up the wage rate to the
level of the expected productivity of the worker, so
firms make 0 profits.
There are two types of workers: low productivity (y1 =
1) and high productivity (y2 = 2). Productivity is
private information of each worker.
Before entering the job market, each worker decides
how much education (e) to get.
Education does not affect productivity or utility of the
worker.
Education is costly:
ct = e/ yt
Job Market Signaling
 After observing the level of e, firms make individual
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wage offers, which can depend on the level of
education.
The goal of the firm is to maximize expected profit.
Workers maximize expected wage.
This game has many equilibria (pooling and
separating)
The following is a separating PBE in this game:
 Type 1 worker chooses e = 0, type 2 worker
chooses e = 1;
 Firms set wages w(e<1) = 1, w(e>=1) = 2
 Firms believe that if e>=1 then type = 2 and if e<1
then type = 1
Limit pricing
(Milgrom-Roberts model)
 Limit pricing is a situation, where na
incumbent monopolist charges a below-cost
price to deter entry of a new firm (or „limit” the
scale or scope of entry)
 The classic rationale for limit pricing, that the
entering firm will get scared of fierce
competition, does not survivie the gametheoretic logic
 Milgrom and Roberts (1982) showed that
what looks like limit pricing could be an
equilibrium in a signaling game
Limit pricing – the game
 Nature decides the incumbent's type. His
marginal cost is low with probability x or high
with prob. (1 – x)
 The incumbent sets the price, observed by
the potential entrant. The entrant may update
her beliefs about the incumbent based on that
price
 Entrant decides whether or not to enter. If no
entry: incumbent enjoys unthreatened
monopoly position. If entry occurs: firms
receive duopoly profits
Limit pricing - notation
 Let i=1 denote the incumbent and i=2 the
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entrant.
Mit(p) - monopoly profit of firm i if her type is t
and charges price p
Mit = max Mit(p) - monopoly profit of firm i if
her type is t and charges the price pmt (the
optimal monopoly price for that type)
Dit - duopoly profit of firm i if incumbent's type
is t
Assume that entrant wants to enter iff type is
high, i.e. D2H > 0 > D2L (example: Bertrand
competition with cH > cE > cL)
Separating equilibria
 Let us look at conditions for and
properties of separating equilibria in this
game
 In a separating equilibrium, the two
types set different prices: pL  pH
 And entry will occur only if the entrant
observes pH. It follows that pH = pmH
Separating equil. cont.
 We have the following constraints on pL:
 ICH: M1H + D1H  M1H(pL) + M1H
or
M1H – M1H(pL)  + (M1H – D1H)
i.e. that the high-cost type will not rather mimic the
low-cost type
 IRL: M1L(pL) + M1L  M1L + D1L
or
M1L – M1L(pL)  + (M1L – D1L)
i.e. that the low-cost type would not rather deviate
and charge pLm
 In equilibrium, the entrant stays out if sees pL
(satisfying the above), enters if sees any price
other than pL (in particular pmH) (beliefs?)
Separating equil. cont.
 It is not easy to prove that separating
equilibria exist, but indeed they do, with pL <
pLm (but not necessarily below cost)
 the incumbent of type L has to give up some
profit in order to discourage entry, the price is
below monopoly price
 social welfare is higher than under perfect
information: entry occurs only when it is
efficient and type L charges a belowmonopoly price in period 1 (limit pricing of this
type is good!)
Pooling equilibria
 In a pooling equilibrium both types charge the
same price pP
 If (1 – x)D2H + xD2L > 0 then the entrant wants
to enter if sees pP, but then at least one of
the types has an incentive to charge a
different price (i.e. Pooling equilibrium must
involve entry deterrence)
 We must therefore have (1 – x)D2H + xD2L < 0
Pooling equil. cont.
 We have the following constraints on pP:
 IRL: M1L(pP) + M1L  M1L + D1L
or
M1L – M1L(pP)  + (M1L – D1L)
i.e. that the low-cost type would not rather deviate
and charge pLm
 IRH: M1H(pP) + M1H  M1H + D1H
or
M1H – M1H(pP)  + (M1H – D1H) i.e. that
the low-cost type would not rather deviate and
charge pHm
 In equilibrium, the entrant stays out if sees pP
(satisfying the above), enters if sees any price
other than pP (beliefs?)
Pooling equil. cont.
 It can be shown that there are many prices
that satisfy the above conditions, but most
importantly, pLm satisfies them (intuitive)
 The low-cost type does not have to worry
about entry, so she chooses her monopoly
price. The H type has to give up some profit
to discourage entry, and charges pLm which is
below her monopoly price
 There is no entry in equilibrium no matter
what the costs are (inefficient)
 Social welfare effect is ambiguous: entry
never occurs, but price sometimes lower
One more thing
 Notice that in a separating equilibrium
the L type is engaged in limit pricing,
while in a pooling equilibrium - the Htype is engaged in limit pricing!
 In both cases it is used as a deterrent
by the type that is most endangered by
entry.
Cooperative game theory
 We do not model how the agents come to an
agreement, we just characterize the players
in terms of their bargaining power and then
look for solutions that satisfy certain desirable
mathematical conditions (axioms)
 Axiomatic bargaining: the bargaining power
is defined by the threat point
 Coalitional Games: slightly more complex,
bargaining power depends on threat points of
every coalition. Coalition formation may be
but does not have to be explicitly modeled or
assumed.
Coalitional games
 Coalitional game with transferable payoffs: the threat
point of any coalition can be represented by a single
number (the value of the coalition). We therefore assume
that what a coalition can achieve can be turned into some
transferable good (money?) and distributed among the
coalition member in an arbitrary manner. The existence of
the value of a coalition is assumed, the usual
interpretation is that what coalition can achieve is
independent of what happens outside of the coalition (if it
isn’t, then non-cooperative GT is more appropriate).
 Formally, the CG with transferable payoffs consists of:
 N - the finite set of players
 v(S) – the (value) function that assigns a real number
to every nonempty set S  N
 Cohesiveness Assumption: v(N) ≥ the sum of values
of any partition of N
The Core
 The Core is the equivalent of NE for coalitional
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games
x(S )  iS xi
Let
The Core of a coalitional game is the set of payoff
profiles x (N payoff vectors) for which x(S) ≥ v(S) for
every S  N
So x is in the core if no coalition can obtain a total
payoff that exceeds the sum of its members’ payoffs
in x
The core is a prediction of the level of transferable
utility (money) that each player will end up with.
Problem (just as with Nash equilibrium): may not be
unique, may not exist.
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