
A Bargaining Model of Domestic Politics and the Cost of War

 Unformatted Document Text:
bargaining game began with the ﬁrst mover proposing a redistribution of the money (higher,
lower, or the same as the initial status quo allocation). The second mover (“responder”)
could either accept the proposal or reject and make a counter oﬀer for dividing the total
allocation, which drops by $1, from $10 to $9 in the second stage.). If a counter oﬀer is
made, the ﬁrst mover can either accept or reject and initiate a conﬂict, where the outcome is
determined by known probabilities and conﬂict costs. In all rounds of our experiments, we
kept the conﬂict costs for each player at $2. The set of feasible demands and counterdemands
was restricted to integer values (0, 1, . . . 10).
The proposer win probabilities were set at either p = 0.2, 0.4, 0.6, or 0.8, with the outcome
explained in terms of a throw of a 10sided die. Each session consisted of 12 subjects being
randomly matched for 10 rounds, with one value of p used in the ﬁrst 5 rounds, and a switch
to another value of p in the ﬁnal 5 rounds. We ran 8 sessions, with treatment sequences of
(.2, .6), (.6, .2), (.2, .8), (.8, .2), (.4, .6), (.6, .4), (.4, .8), and (.8, .4). Thus we ran each
possible combination with a low win probability followed by a high probability, or vice versa.
The 96 subjects were University of Virginia students who were paid $6 for participating, plus
half of accumulated earnings. Earnings were in the $18$25 range, for a onehour session,
and were paid immediately after each session.
The quantal response probabilities of initial proposer demands can be used to calculate
the expected value of the initial demand, as a function of the four proposer win probabilities.
With a high rationality parameter of λ = λ
P
= λ
R
= 10, the predicted initial demands are
1.00, 3.04, 5.07, and 7.03, which essentially match the Nash predictions of 1, 3, 5, and 7 that
are easily obtained with standard backward induction arguments.
4
These Nash predictions
are connected by the dashed line on the left side of Figure 1, and the Nash zeroconﬂict
4
The Nash predictions can be veriﬁed directly, under the assumption that a responder will accept an oﬀer
that makes the responder indiﬀerent, as assumed in the onestage bargaining model. For example, if theproposer win probability is 0.8, the proposer’s expected gain in the second stage is (0.8)*9 minus the conﬂictcost of 2, which yields 7.22 = 5.2. The responder could avoid the conﬂict by making the next higher integeroﬀer of 6 to the proposer and keeping the remaining amount, 3, from the secondstage pie. In the ﬁrst stage,the proposer can therefore, oﬀer the responder 3 and keep 7, which is the Nash ﬁrststage demand, assumingthat the responder accepts in the case of indiﬀerence. The other Nash predictions can be veriﬁed in a similarmanner.
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 Authors: Clark, David., Holt, Charles., Nordstrom, Timothy., Reed, William. and Sieberg, Katri. 




bargaining game began with the ﬁrst mover proposing a redistribution of the money (higher,
lower, or the same as the initial status quo allocation). The second mover (“responder”)
could either accept the proposal or reject and make a counter oﬀer for dividing the total
allocation, which drops by $1, from $10 to $9 in the second stage.). If a counter oﬀer is
made, the ﬁrst mover can either accept or reject and initiate a conﬂict, where the outcome is
determined by known probabilities and conﬂict costs. In all rounds of our experiments, we
kept the conﬂict costs for each player at $2. The set of feasible demands and counterdemands
was restricted to integer values (0, 1, . . . 10).
The proposer win probabilities were set at either p = 0.2, 0.4, 0.6, or 0.8, with the outcome
explained in terms of a throw of a 10sided die. Each session consisted of 12 subjects being
randomly matched for 10 rounds, with one value of p used in the ﬁrst 5 rounds, and a switch
to another value of p in the ﬁnal 5 rounds. We ran 8 sessions, with treatment sequences of
(.2, .6), (.6, .2), (.2, .8), (.8, .2), (.4, .6), (.6, .4), (.4, .8), and (.8, .4). Thus we ran each
possible combination with a low win probability followed by a high probability, or vice versa.
The 96 subjects were University of Virginia students who were paid $6 for participating, plus
half of accumulated earnings. Earnings were in the $18$25 range, for a onehour session,
and were paid immediately after each session.
The quantal response probabilities of initial proposer demands can be used to calculate
the expected value of the initial demand, as a function of the four proposer win probabilities.
With a high rationality parameter of λ = λ
P
= λ
R
= 10, the predicted initial demands are
1.00, 3.04, 5.07, and 7.03, which essentially match the Nash predictions of 1, 3, 5, and 7 that
are easily obtained with standard backward induction arguments.
4
These Nash predictions
are connected by the dashed line on the left side of Figure 1, and the Nash zeroconﬂict
4
The Nash predictions can be veriﬁed directly, under the assumption that a responder will accept an oﬀer
that makes the responder indiﬀerent, as assumed in the onestage bargaining model. For example, if the proposer win probability is 0.8, the proposer’s expected gain in the second stage is (0.8)*9 minus the conﬂict cost of 2, which yields 7.22 = 5.2. The responder could avoid the conﬂict by making the next higher integer oﬀer of 6 to the proposer and keeping the remaining amount, 3, from the secondstage pie. In the ﬁrst stage, the proposer can therefore, oﬀer the responder 3 and keep 7, which is the Nash ﬁrststage demand, assuming that the responder accepts in the case of indiﬀerence. The other Nash predictions can be veriﬁed in a similar manner.
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