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