What I mean is that you can think of “CDT agent with certain utility function” and “FDT agent” as exactly the same. They’re the same concept.
They are not. A CDT agent is fundamentally doing a different expected value calculation than a FDT agent. This is why they can lead to radically different outcomes.
I should have clarified more, oops. I was talking about a minor variation of the scenario where the “negotiation is not possible” restriction is lifted (while still keeping the information asymmetry somehow).
Okay, play out the scenario. He offers to take CDT will back for $199.99, what does will say? Will’s expected values are:
“Ok”—payoff = $0.01
“No”—payoff = -$1,000,000 (this is assuming an honest/total no, the other case is under 3)
“No take me for X amount.”—payoff = ???[1] (he doesn’t know whether Derek will accept or not, note that this includes the case where X is zero or the cases where X is negative).
Now the question becomes “how does Will estimate the payoffs for 3?” What is his expectation for Derek to negotiate? Etc. If we assume sufficient risk aversion (which I would argue is the most probable outcome) 1 is still preferable.
Let’s imagine Derek offers to take FDT will back for $1,000,199.99. Will’s expected values are:
Be an agent that would say “OK”—payoff = -$999,999.99
Be an agent that would say “No”—payoff = -$1,000,000
Be an agent that would say “no take me for X amount”—payoff = ???
Edit: Be an agent that would say “Ok” but not actually pay Derek—payoff = -$1,000,000 (I realize I forgot to include this one, as in Parfit’s Hitchiker, the agents’ expected outcome if they wouldn’t pay Derek honestly is that they would be left to die)[2]
FDT-Will has the same problem as CDT-Will. Though, for FDT-Will, unlike CDT-Will, I would argue under most reasonable assumptions there would be some preferable value for X under 3 that FDT-Will would estimate has a better expected value. Given that, he would try to negotiate a value somewhere between -$1 and $1,000,199.99. Where he would negotiate that value depends on risk aversions and how he estimates the responses from Derek.
And meanwhile if Derek’s $1,000,000 value on honesty is set to $0 BUT he uses FDT then the exact same thing happens absent any weird commitment-race dynamics with FDT-Will
I am not convinced this is true. I don’t see why FDT-Derek would behave differently. If we assume information symmetry, then you get the same commitment race with their priors.
FDT-Will then says “give me $0.99 and I’ll let you save my life”
Why? See above. FDT-Will under your scenario still suffers from information asymmetry. You can argue that the 3 is reasonably the better option for him, but he has no idea what value of X is optimal. We know Derek considers any value >-$0.99 as an expected positive, but Will is operating in an asymmetric environment. He doesn’t know what Derek will decide. It seems reasonable that Will might expect Derek to accept some lower amount, but he is going to have to weigh that against the probability that Derek says “no.” If he has extreme risk aversion, he will still prefer 1 even if he estimates Derek would likely accept a lower price. If he has no risk aversion and Derek cannot counter offer, he will offer whatever he expects Derek to accept.
and poor CDT-Derek will agree.
Why? Let’s lay out CDT-Derek’s option.
Agree—payoff = -$5.99
Refuse—payoff = -$6.00
Refuse and tell FDT-Will “I will only take you back for X amount” (where X is greater than −0.99) - payoff = unspecified (but known to Derek)
It seems likely that CDT-Derek would pick some variant of 3, dependent on what he expects FDT-Will to react with which depends on FDT-Will’s estimation of CDT-Derek. You would expect to get some race with FDT-Will trying to determine Derek’s utility function. Indeed, if we assume perfect information asymmetry, CDT-Derek’s best move is probably to keep saying “I will only take you back for $1,000,199.99” to prevent FDT-Will from getting any information on his utility, if CDT-Derek repeats that until FDT-Will is about to die if he doesn’t make a decision, FDT-Will, having gained no information on the Derek’s utility, is likely to simply accept when he becomes unable to negotiate further (for the same reasons as above).[3] And, making the standard FDT estimates when he gets to town (i.e., he anticipates if he wasn’t the kind of agent that would pay, he would have been left to die), he would honestly pay the $1,000,199.99.
When I use ‘???’ I mean that it is both unspecified under the assumptions of the equations and unknown to the agent. We would need to add additional specifications to the problem to determine the expected payoff for different values of X, and without changing assumptions Will’s expected payoff from X would remain unknown to Will. We could add assumptions for Will’s estimates (which are not likely to be equivalent to the real payoffs), to determine what Will would estimate the expected payoffs are for different values of X.
I am not including this option for the CDT agent since it is a strictly inferior version of 1, since their payout for honesty is $200 it is trivial that they would always be honest for under $199.99
If there is no such cutoff, they are in a classic battle of the sexes type problem FDT-Will’s expected payoff from a deal is $1,000,200 - X, where Derek’s payoff is $1 + X. It is not clear what FDT-Will’s position would have to be for him to expect CDT-Derek to accept a better deal. Any deal from −0.99 to $1,000,199.99 is feasible under our assumptions (and would be a Nash Equilibrium) but we have no reason to expect any outcome in that range without adding some assumptions.
I don’t have the time to give this full consideration, but on the whole I think you are correct if Will has the information asymmetry in both the negotiation phase and the payment phase, whereas I was implicitly assuming Will having full information in negotiation and suddenly gaining an information asymmetry in the payment phase (which doesn’t make much sense). So, yeah, I think I agree.
They are not. A CDT agent is fundamentally doing a different expected value calculation than a FDT agent. This is why they can lead to radically different outcomes.
Okay, play out the scenario. He offers to take CDT will back for $199.99, what does will say? Will’s expected values are:
“Ok”—payoff = $0.01
“No”—payoff = -$1,000,000 (this is assuming an honest/total no, the other case is under 3)
“No take me for X amount.”—payoff = ???[1] (he doesn’t know whether Derek will accept or not, note that this includes the case where X is zero or the cases where X is negative).
Now the question becomes “how does Will estimate the payoffs for 3?” What is his expectation for Derek to negotiate? Etc. If we assume sufficient risk aversion (which I would argue is the most probable outcome) 1 is still preferable.
Let’s imagine Derek offers to take FDT will back for $1,000,199.99. Will’s expected values are:
Be an agent that would say “OK”—payoff = -$999,999.99
Be an agent that would say “No”—payoff = -$1,000,000
Be an agent that would say “no take me for X amount”—payoff = ???
Edit: Be an agent that would say “Ok” but not actually pay Derek—payoff = -$1,000,000 (I realize I forgot to include this one, as in Parfit’s Hitchiker, the agents’ expected outcome if they wouldn’t pay Derek honestly is that they would be left to die)[2]
FDT-Will has the same problem as CDT-Will. Though, for FDT-Will, unlike CDT-Will, I would argue under most reasonable assumptions there would be some preferable value for X under 3 that FDT-Will would estimate has a better expected value. Given that, he would try to negotiate a value somewhere between -$1 and $1,000,199.99. Where he would negotiate that value depends on risk aversions and how he estimates the responses from Derek.
I am not convinced this is true. I don’t see why FDT-Derek would behave differently. If we assume information symmetry, then you get the same commitment race with their priors.
Why? See above. FDT-Will under your scenario still suffers from information asymmetry. You can argue that the 3 is reasonably the better option for him, but he has no idea what value of X is optimal. We know Derek considers any value >-$0.99 as an expected positive, but Will is operating in an asymmetric environment. He doesn’t know what Derek will decide. It seems reasonable that Will might expect Derek to accept some lower amount, but he is going to have to weigh that against the probability that Derek says “no.” If he has extreme risk aversion, he will still prefer 1 even if he estimates Derek would likely accept a lower price. If he has no risk aversion and Derek cannot counter offer, he will offer whatever he expects Derek to accept.
Why? Let’s lay out CDT-Derek’s option.
Agree—payoff = -$5.99
Refuse—payoff = -$6.00
Refuse and tell FDT-Will “I will only take you back for X amount” (where X is greater than −0.99) - payoff = unspecified (but known to Derek)
It seems likely that CDT-Derek would pick some variant of 3, dependent on what he expects FDT-Will to react with which depends on FDT-Will’s estimation of CDT-Derek. You would expect to get some race with FDT-Will trying to determine Derek’s utility function. Indeed, if we assume perfect information asymmetry, CDT-Derek’s best move is probably to keep saying “I will only take you back for $1,000,199.99” to prevent FDT-Will from getting any information on his utility, if CDT-Derek repeats that until FDT-Will is about to die if he doesn’t make a decision, FDT-Will, having gained no information on the Derek’s utility, is likely to simply accept when he becomes unable to negotiate further (for the same reasons as above).[3] And, making the standard FDT estimates when he gets to town (i.e., he anticipates if he wasn’t the kind of agent that would pay, he would have been left to die), he would honestly pay the $1,000,199.99.
When I use ‘???’ I mean that it is both unspecified under the assumptions of the equations and unknown to the agent. We would need to add additional specifications to the problem to determine the expected payoff for different values of X, and without changing assumptions Will’s expected payoff from X would remain unknown to Will. We could add assumptions for Will’s estimates (which are not likely to be equivalent to the real payoffs), to determine what Will would estimate the expected payoffs are for different values of X.
I am not including this option for the CDT agent since it is a strictly inferior version of 1, since their payout for honesty is $200 it is trivial that they would always be honest for under $199.99
If there is no such cutoff, they are in a classic battle of the sexes type problem FDT-Will’s expected payoff from a deal is $1,000,200 - X, where Derek’s payoff is $1 + X. It is not clear what FDT-Will’s position would have to be for him to expect CDT-Derek to accept a better deal. Any deal from −0.99 to $1,000,199.99 is feasible under our assumptions (and would be a Nash Equilibrium) but we have no reason to expect any outcome in that range without adding some assumptions.
I don’t have the time to give this full consideration, but on the whole I think you are correct if Will has the information asymmetry in both the negotiation phase and the payment phase, whereas I was implicitly assuming Will having full information in negotiation and suddenly gaining an information asymmetry in the payment phase (which doesn’t make much sense). So, yeah, I think I agree.