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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 counter-demands 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 10-sided 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 one-hour 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 zero-conflict 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 one-stage 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 7.2-2 = 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 second-stage pie. In the first stage,the proposer can therefore, offer the responder 3 and keep 7, which is the Nash first-stage demand, assumingthat the responder accepts in the case of indifference. The other Nash predictions can be verified in a similarmanner. 13

Authors: Clark, David., Holt, Charles., Nordstrom, Timothy., Reed, William. and Sieberg, Katri.
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background image
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 counter-demands
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 10-sided 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 one-hour 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 zero-conflict
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 one-stage 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 7.2-2 = 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 second-stage pie. In the first stage,
the proposer can therefore, offer the responder 3 and keep 7, which is the Nash first-stage demand, assuming
that the responder accepts in the case of indifference. The other Nash predictions can be verified in a similar
manner.
13


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