# Signaling games

Signaling games
• consider two firms
– Oldstar ( old , set in the market)
– Nova (new)
• if fight happens ,oldstar can beat weak
nova but not the strong, the winner has the
market to itself.
• for oldstar 3 ,for nova 4 and cost of fighting
is -2
• payoff matrix
fight
refrain
strong
2,-2
4,0
weak
-2,1
0,3
equilibrium without signaling
• let ‘w’ be probability that nova is weak
• in absence of any signals from nova, the
payoff function for fighting is
(w)1+(1-w)(-2) &gt;0
w &gt; 2/3
so oldstar fights if it has a prior information
that nova is weak
signaling
• nova can give information by
– display
– don’t display
• strong nova
fight
refrain
challen 2,-2
ge
4,0
don't
0,3
challen
ge
0,3
• weak nova , c is the
cost for displaying
fight
refrain
challen -2-c,1
ge and
display
2-c,0
challen -2,1
ge but
don't
display
2,0
• so if w &lt;2/3 , then oldstar retreats if it
sees the display.
• so for weak nova , if c &lt;2 then it should
challenge and display, since oldstar
retreats – pooling equilibrium
how does oldstar react
• Oldstar draws
conclusion whether or
not nova displays
according to Bays
rule
displ no
ay
displ
ay
stron 1-w 0
g
wea
k
wp
sum
of
col
1-w
w(1- w
p)
sum 1w(1of
w+w p)
row p
semi - separation
• so Old stars payoff from fighting
conditional on oberving a display is
1(wp/(1-w+wp)) + (-2)(1-w)/(1-w+wp)
= [wp – 2(1-w)]/(1-w+wp)
• nova chooses p to keep oldstar perfectly
indifferent
p = 2(1-w)/w
Mixed strategy
• Old stars strategy of fighting q , weak
nova’s expected payoff form challenging a
display
q(-2-c) + (1-q)(2-c) = 2-c-4q
• weak nova’s payoff for not challenging = 0
•
q = (2-c)/4