Also, putting this in another post since I think it is a major point, if we assume some cost to bargaining, for Derek it approximates something like a dove-hawk game, where Derek gets the first move. Will’s game is more complex as he is operating under information assymetry, so depends on the odds he assigns some probailities
If we consider the value Will pays as X (negative if Derek pays Will), if we assume some cost (C) of both negotiating the outcome, the payoffs works out to (I don’t know how/if you can put tables into comments so I just have to write them out):
Payoffs given FDT-Will with Negotiation:
(1) Will accepts the initial offer (for FDT-Will, X = 1,000,199.99):
Derek: 1 + 1,000,199.99
Will: 0 * −1,000,000 + 1 * −999,999.99
(2) Will Contests and Derek Accepts (say X = −0.99[1]):
(3) Will and Derek contest over X. X is unspecified under the assumptions, any number where X > (C − 1) and X < (200 - C) is feasible:
Derek: 1 + X—C
Will: P(Derek rejects)*-1,000,0000 + P(Derek accepts)*(200- X) - C
Counterfactual: Derek doesn’t offer an amount and Will doesn’t contest (X = 0)
Derek: 1
Will: 1,000,200
While we would need to know Will’s probability estimates to actually model how they behave and what actions they take, from this it seems rather evident that under most approximations CDT-Will is still likely to be better off.
Also, putting this in another post since I think it is a major point, if we assume some cost to bargaining, for Derek it approximates something like a dove-hawk game, where Derek gets the first move. Will’s game is more complex as he is operating under information assymetry, so depends on the odds he assigns some probailities
If we consider the value Will pays as X (negative if Derek pays Will), if we assume some cost (C) of both negotiating the outcome, the payoffs works out to (I don’t know how/if you can put tables into comments so I just have to write them out):
Payoffs given FDT-Will with Negotiation:
(1) Will accepts the initial offer (for FDT-Will, X = 1,000,199.99):
Derek: 1 + 1,000,199.99
Will: 0 * −1,000,000 + 1 * −999,999.99
(2) Will Contests and Derek Accepts (say X = −0.99[1]):
Derek: 1 − 0.99
Will: P(Derek rejects)*-1,000,0000 + P(Derek accepts)*(200.99) + P(Derek contests)*( (3) Will)
(3) Will and Derek contest over X. X is unspecified under the assumptions, any number where X > (C − 1) and X < (1,000,200 - C) is feasible:
Derek: 1 + X—C
Will: P(Derek rejects)[2]*-1,000,0000 + P(Derek accepts)*(200- X) - C
Counterfactual: Derek doesn’t offer an amount and Will doesn’t contest (X = 0)
Derek: 1
Will: 1,000,200
Payoffs given CDT-Will with Negotiation:
(1) Will accepts the initial offer (for CDT-Will, X = 199.99, since anything greater wouldn’t be paid):
Derek: 1 + 199.99
Will: 0 * −1,000,000 + 1 * 0.01
(2) Will Contests and Derek Accepts (X = −0.99):
Derek: 1 − 0.99
Will: P(Derek rejects)*-1,000,0000 + P(Derek accepts)*(200.99) + P(Derek contests)*( (3).Will)
(3) Will and Derek contest over X. X is unspecified under the assumptions, any number where X > (C − 1) and X < (200 - C) is feasible:
Derek: 1 + X—C
Will: P(Derek rejects)*-1,000,0000 + P(Derek accepts)*(200- X) - C
Counterfactual: Derek doesn’t offer an amount and Will doesn’t contest (X = 0)
Derek: 1
Will: 1,000,200
While we would need to know Will’s probability estimates to actually model how they behave and what actions they take, from this it seems rather evident that under most approximations CDT-Will is still likely to be better off.
Realistically, Will could set a value of X higher to decrease the chances of Derek contesting. But I am just assuming the extreme case here.
These probabilities depend on the values of X. FDT-Will would estimate that as x approaches 1,000,199.99 P(Derek accepts) approaches 1.