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A Bargaining Model of Domestic Politics and the Cost of War
Unformatted Document Text:  demand x 2 times the probability that it is accepted (first term in square brackets below) plus the responder’s expected payoff from conflict, δ(1 − p) − c R , times the probability of rejection (second term in square brackets below): π R (x 2 ) = x 2 e (δ−x 2 )λ P e (δ−x 2 )λ P + e (δp−c P )λ P ) + (δ(1 − p) − c R ) e (δp−c P )λ P e (δ−x 2 )λ P + e (δp−c P )λ P ) (1) Notice that the probabilities of acceptance and rejection are ratios of exponentials of the proposer’s payoffs for acceptance and conflict, and that these probabilities depend on λ P , the proposer’s rationality parameter. Next, the probability that the responder will select any counter-demand x 2 is a ratio of exponentials of the expected payoffs for various counter- demands, each multiplied by the responder’s rationality parameter: P r(x i ) = e π R (x 2 )λ R j e π R (x 2j )λ R (2) where the sum in the denominator is taken over the various possible counter-demands that the responder may consider. The responder’s expected payoff (REP 2 ) for the second stage, then is the summation of the expected payoffs for each x 2 , multiplied by the probability that the responder demands that level of x 2 : REP 2 = π R (x 2 )P r(x 2 ). Similarly, the proposer’s expected payoff for a given counter-demand x 2 received is the proposer’s share, δ − x 2 , times the proposer’s own acceptance probability, plus the proposer’s conflict payoff, δp − c P , times the conflict probability: π P (x 2 ) = (δ − x 2 ) e (δ−x 2 )λ P e (δ−x 2 )λ P + e (δp−c P )λ P + (δp − c P ) e (δp−c P )λ P e (δ−x 2 )λ P + e (δp−c P )λ P (3) 11

Authors: Clark, David., Holt, Charles., Nordstrom, Timothy., Reed, William. and Sieberg, Katri.
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background image
demand x
2
times the probability that it is accepted (first term in square brackets below)
plus the responder’s expected payoff from conflict, δ(1 − p) − c
R
, times the probability of
rejection (second term in square brackets below):
π
R
(x
2
) = x
2
e
(δ−x
2
P
e
(δ−x
2
P
+ e
(δp−c
P
P
)
+ (δ(1 − p) − c
R
)
e
(δp−c
P
P
e
(δ−x
2
P
+ e
(δp−c
P
P
)
(1)
Notice that the probabilities of acceptance and rejection are ratios of exponentials of the
proposer’s payoffs for acceptance and conflict, and that these probabilities depend on λ
P
,
the proposer’s rationality parameter. Next, the probability that the responder will select
any counter-demand x
2
is a ratio of exponentials of the expected payoffs for various counter-
demands, each multiplied by the responder’s rationality parameter:
P r(x
i
) =
e
π
R
(x
2
R
j
e
π
R
(x
2j
R
(2)
where the sum in the denominator is taken over the various possible counter-demands that
the responder may consider. The responder’s expected payoff (REP
2
) for the second stage,
then is the summation of the expected payoffs for each x
2
, multiplied by the probability that
the responder demands that level of x
2
: REP
2
=
π
R
(x
2
)P r(x
2
).
Similarly, the proposer’s expected payoff for a given counter-demand x
2
received is the
proposer’s share, δ − x
2
, times the proposer’s own acceptance probability, plus the proposer’s
conflict payoff, δp − c
P
, times the conflict probability:
π
P
(x
2
) = (δ − x
2
)
e
(δ−x
2
P
e
(δ−x
2
P
+ e
(δp−c
P
P
+ (δp − c
P
)
e
(δp−c
P
P
e
(δ−x
2
P
+ e
(δp−c
P
P
(3)
11


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