Download Salop Model of Product Differentiation Consumers are located

Salop Model of Product Differentiation
Consumers are located uniformly on a circle
(we normalize its perimeter to 1) and have
linear transport costs, T (d) = td. A large
number of firms play the following game.
In the first stage, they simultaneously decide whether to enter. Those that do get
in, are distributed evenly along the circle,
maximizing the amount of differentiation.
1
Once in, entrant firms make profits (pi −
c)Di − f , where f is the fixed cost of entry
and Di is firm i’s demand.
Suppose that n firms enter.
We look for
a symmetric equilibrium in which all firms
charge the same price p.
Consider now the decision of firm i given
that the rest choose p.
2
By setting a price pi, its demand is Di(pi, p) =
2x, where x(0, 1/n) is the location of the
consumer that is indifferent between purchasing from i or its neighbour:
1 − x) ⇒ x = 1 p−pi +t/n
pi + tx = p + t( n
2
t
Profit maximization for firm i yields pi =
c + nt . This must then be the equilibrium
price.
3
With n firms entering, the equilibrium profits are nt2 − f , which are decreasing in the
number of firms.
With free entry, n is determined by the zeroprofit condition: n =
q
t/f . Notice that at
the equilibrium level of entry, the price set
in stage two, p = c +
√
tf is above marginal
cost.
4
Example: Dominance in mixed strategies
Consider the 2-player normal form game depicted below:
Player 1
U
M
D
Player 2
L
R
10,1 0,4
4,2
4,3
0,5 10,2
No strict dominance in pure strategies. But
if player 1 plays U and D with probability 1
2
each, he has an expected payoff of 5 regardless of player 20s strategy, and this strategy
strictly dominates M
5
Correlated equilibria
So far, we have assumed that players’ randomisations (when they play mixed strategies) are independent. In the 2×2 coordination games considered earlier, for instance,
we can describe the mixed strategy equilibrium as follows:
Nature provides private and independently
distributed signals (θ1, θ2) ∈ [0, 1] × [0, 1] to
the 2 players, and each player i assigns decisions to various possible realisations of θi.
6
Suppose there are also public signals available that both players observe. Let θ ∈ [0, 1]
be one such signal. Suppose both decide to
1 and to the game if
go the dance if θ ≤ 2
θ > 1
2 (in the ‘battle of partners’ coordination game).
Each player’s strategy choice is still random
but now their actions are perfectly coordinated, and they always meet.
7
Moreover, these decisions have an equilibrium character; if 1 player follows this decision rule, it is optimal for the other player
also to do so.
This kind of equilibrium is called a ‘correlated equilibrium’ (due to Aumann) - related
to the concept of ‘sunspots’ in Macroeconomics.
8
Proposition: Every game ΓN = [I, {∆(Si)}, {ui(.)}]
in which the sets S1, ..., SI have a finite number of elements has a mixed strategy NE
(existence)
Proposition: A NE exists in game ΓN =
[I, {Si}, {ui(.)}], if for all i = 1, ..., I,
(i) Si is a nonempty, convex and compact
subset of some Euclidean space RM ;
(ii) ui(s1, ..., sI ) is continuous in (s1, ..., sI )
and quasiconcave in si.
9
Static games of incomplete information
- Bayesian games
In a Bayesian game, each player i has a payoff function ui(si, s−i, θi), where θi ∈ Θi is a
random variable chosen by nature that is
observed only by player i.
Joint probability distribution of the θis is
given by F (θ1, ...θI ) - common knowledge.
Letting Θ = Θ1 × ... × ΘI , a Bayesian game
is summarised by [I, {Si}, {ui}, Θ, F (.)]
10
A pure strategy in a Bayesian game for player
i is a function si(θi) that gives the player’s
strategy choice for each realisation of his
type θi. Player i0s pure strategy set Si is the
set of all such functions.
Player i0s expected payoff for a given profile
of pure strategies for the I players (s1(.), ..., sI (.))
is then given by:
ũi(si(.), ....sI (.)) = Eθ [ui(s1(θ1), ...sI (θI ), θi)]
11
Definition: A (pure strategy) Bayesian Nash
equilibrium (BNE) for the Bayesian game
[I, {Si}, {ui}, Θ, F (.)] is a profile of decision
rules (s1(.), ..., sI (.)) that constitutes a Nash
equlibrium of game ΓN = [I, {Si}, {ũi}]. That
is, for every i = 1, ...I,
ũi(si(.), s−i(.)) ≥ ũi(s0i(.), s−i(.))
∀s0i(.) ∈ Si.
12
Proposition: A profile of decision rules (s1(.), ...sI (.))
is a BNE for the Bayesian game [I, {Si}, {ui}, Θ, F (.)]
iff, ∀ i and ∀ θ¯i ∈ Θi occurring with positive
probability,
Eθ−i [ui(si(θ¯i), s−i(θ−i), θ¯i|θ¯i] ≥
Eθ−i [ui(s0i, s−i(θ−i), θ¯i|θ¯i]
∀ s0i ∈ Si, where the expectation is taken
over realisations of the other players’ random variables, conditional on player i0s realisation of his signal θ¯i.
13
i.e., we can think of each type of player i
as being a separate player who maximises
his payoff given his conditional probability
distribution over the strategy choices of his
rivals.
Player 1
Player 1
Don’t confess
Confess
Player 2 (µ)
Don’t
Confess
confess
0,-2
-10,-1
-1,-10
-5,-5
Don’t confess
Confess
Player 2 (1 − µ)
Don’t
Confess
confess
0,-2
-10,-6
-1,-10
-5,-11
With probability µ, prisoner 2 has preferences as in 1st matrix (Type I), and with
probability (1 − µ), he’s the type who hates
to snitch - he gets a psychic penalty equal
to 6 years in prison for confessing (Type II).
14
Player 2 now has 4 possible pure strategies:
(confess if type I, confess if type II);
(confess if type I, don’t confess if type II);
(don’t confess if type I, confess if type II);
(don’t confess if type I, don’t confess if type
II)
Player 1 does not observe player 2’s type,
so a pure strategy for player 1 is simple
{confess, don’t confess}
15
To solve for the BNE of the game, note
that type I player must play ‘confess’ with
probability 1 - that is his dominant strategy.
Likewise, type II has a dominant strategy ‘don’t confess’.
Given this, player 10s best response is to play
1 and to play ‘confess’
‘don’t confess’ if µ < 6
1.
.
He’s
indifferent
if
µ
=
if µ > 1
6
6
16
First-price sealed bid auction
We have 2 bidders i = 1, 2. Bidder i has
valuation vi for the good. The 2 bidders’
valuations are independently and uniformly
distributed on [0, 1]. Bids bi ≥ 0. Bidders
simultaneously submit bids. Higher bidder
wins the good and pays the price she bid;
other bidder gets nothing. In case of a tie,
winner determined by flip of a coin. Bidders are risk-neutral. All of this is common
knowledge.
17
Here, Ai = [0, ∞); Θ = [0, 1] × [0, 1];



 vi − bi, bi
0, bi <
ui(b1, b2; v1, v2) =


 (vi−bi) , b
i
2
> bj
bj
= bj
A strategy for player i is a function bi(vi).
In a BNE, player 10s strategy b1(v1) is a BR
to player 20s strategy b2(v2), and vice versa.
The pair of strategies (b(v1), b(v2)) is a BNE
if for each bi in [0, 1], bi(vi) solves:
1 (v −b )P r{b =
maxbi (vi−bi)P r{bi > bj (vj )}+ 2
i
i
i
bj (vj )}
18
We simplify the problem by looking for a
linear equilibrium α1 + β1v1 and α2 + β2v2.
Suppose player j adopts the strategy αj +
βj vj . For given vi, player i0s BR solves:
maxbi (vi − bi)P r(bi > αj + βj vj )
We have αj ≤ bi ≤ αj + βj , for a sensible
strategy!
So, P r(bi > αj + βj vj ) = P r{vj <
bi −αj
=
βj
bi −αj
βj }
19
Player i0s BR is therefore:

vi +αj


 2 , v i ≥ αj
bi(vi) =


 α ,v < α
j i
j
Since we are looking for a linear equilibrium,
we rule out cases where 0 < αj < 1, looking
instead at ranges where αj ≤ 0 and αj ≥ 1.
But αj ≥ 1 cannot be an equilibrium - since
it is optimal for a higher type to bid at least
as much as a lower type’s optimal bid, we
have cj ≥ 0.
20
But then αj ≥ 1 would imply bj (vj ) ≥ vj ,
which cannot be optimal.
So, for a linear equilibrium, we must have
αj ≤ 0, which ⇔ bi(vi) =
vi +αj
2 , so that
α
αi = 2j and ci = 1
2.
Using symmetry to derive j 0s BR, and combining the results, we get αi = αj = 0 and
1.
βi = βj = 2
v
The BNE of this game is { v2i , 2j }
21