In the Ultimatum bargaining game, the experimenter sets up two subjects to play a game only once.
The experimenter offers $10, to be divided between the subjects.
One player, the Proposer, offers a split to the other player, the Responder.
The Responder can either accept, in which case both parties get the Proposer's chosen split; or else refuse, in which case both the Responder and Proposer receive nothing.
What is the minimum offer a rational Responder should accept? What should a rational Proposer offer a rational Responder?
The Ultimatum Game stands in for the problem of dividing gains from trade in non-liquid markets. Suppose:
In principle the trade could take place at any price between $5001 and $7999. In the former case, I've only gained $1 of value from the trade; in the latter case, I've gained $2999 of the $3000 of surplus value generated by the trade. If we can't agree on a price, no trade occurs and no surplus is generated at all.
Unlike the laboratory Ultimatum Game, the used-car scenario could potentially involve reputation effects, and multiple offers and counteroffers. But there's a sense in which the skeleton of the problem closely resembles the Ultimatum Game: if the car-buyer seems credibly stubborn about not offering any more than $5300, then this is the equivalent of the Proposer offering $1 of $10. I can either accept 10% of the surplus value being generated by the trade, or both of us get nothing.
In situations like unions at the bargaining table (with the power to shut down a company and halt further generation of value) or non-monetary trades, the lack of any common market and competing offers makes the Ultimatum Game a better metaphor still.
A large experimental literature exists on the Ultimatum Game and its variants. The general finding is that human subjects playing the Ultimatum Game almost always accept offers of $5, and reject lower offers with increasing probability.
One paper lists acceptance rates as follows:
Among participants with a high score on the Cognitive Reflection Test, the graph looks like it says:
On the academically standard causal decision theory, a rational Responder should accept $1 (or higher), since the causal result of accepting the $1 offer is being $1 richer than the causal result if you reject the $1. Thus, a rational Proposer should propose $1 if it knows it is facing a rational Responder. The much lower acceptance rates by human subjects for $1 offers is therefore evidence of human irrationality.
As in the Transparent Newcomb's Problem, by the time the $1 offer is received, the EDT agent thinks it is too late for its decision to be news about the probability of receiving a $1 offer. Thus, rejecting the $1 offer would be bad news, and the EDT agent will accept the $1. Thus, among two EDT agents with common knowledge of each other's EDT-rationality, the Proposer will offer $1 and the Responder will accept $1.
Under logical decision theories the Ultimatum Game has the potential to be much more complicated. The Responder can take into account that the Proposer may have reasoned about the Responder's output in the course of deciding which offer to send, meaning that the Responder-algorithm's logical reply to different offers can affect both whether the Responder gets the money, and also affect how much money is offered in the first place. Under updateless forms of LDT, there is no obstacle to the Responder taking into account both effects even under the supposition that the Responder has already observed the Proposer's offer.
Suppose the Proposer and Responder have exact common knowledge of each other's algorithms. If an LDT Proposer is facing a CDT/EDT Responder, obviously an LDT Proposer will offer $1 and the CDT/EDT Responder will take it.
Next suppose that a pure EDT or CDT Proposer is facing an updateless LDT Responder, with common knowledge of code. Again, the outcome seems straightforward: the EDT/CDT Proposer will run a simulation of the LDT agent facing every possible offer from $1 to $9, will discover that the simulated LDT agent "irrationally" and very predictably rejects any offer less than $9. So the EDT/CDT Proposer will, with a sigh, give up and offer the LDT agent $9. [3]
The case where two LDT agents face each other seems considerably more complicated. If one were to try to gloss LDT as the rule "Do what you would have precommitted to do", then the situation seems to dissolve into precommitment warfare in which neither side has any obvious advantage or way to precommit "logically first".
No formal solution to this problem has been proven from scratch. Informally, Eliezer Yudkowsky has suggested (in a workshop discussion) that an LDT equilibrium might arise as follows:
Suppose the LDT Responder thinks that a 'fair' solution ('fair' being a term of art in the process of definition) is $5 apiece for Proposer and Responder, e.g. because $5 is the Shapley value.
However, the Responder is not sure that the LDT Proposer considers Shapley division to be the 'fair' division of the gains. Then:
This suggests that an LDT Responder should definitely accept any offer at or above what the Responder thinks is the 'fair' amount of $5, and probabilistically accept or reject any offer below $5, such that the expected value of the Proposer slopes downward as that agent proposes lower splits. E.g., the Responder might accept an offer beneath $5 with probability:
$$p = \big ( \frac{$5}{$10 - q} \big ) ^ {1.01}
This implies that, e.g., an offer of $4 might be accepted with 83% probability, implying an expected gain to the Proposer of $4.98, and an expected gain to the Responder of $3.32.
From the Responder's perspective, the important feature of this response function is not that the Responder gets the same amount as the Proposer. Rather, the key feature is that the Proposer gains no expected benefit from giving the Responder less than the 'fair' amount (and further-diminished returns as the Responder's returns decrease further).
What about the LDT Proposer? It does not want to be the sort of agent which, faced with another agent that accepts only offers of $9, gives in and offers $9. Faced with an agent that is or might be like that, one avenue might be to simply offer $5, and another avenue might be to offer $9 with 50% probability and $1 otherwise (leading to an expected Responder gain of $4.50). No Responder should be able to do better by coming in with a notion of 'fairness' that demands more than what the Proposer thinks is 'fair'.
This gives the notion of 'fairness'--even if it so far seems arbitrary in terms of fundamentals--a very Schelling Point-like status. From the perspective of both agents:
This solution is not on the Pareto boundary. In fact, Yudkowsky proposed it in response to an earlier proof by Stuart Armstrong that it was impossible to develop a bargaining solution with certain properties that did lie on the Pareto boundary. (Specifically, there was provably no solution on the Pareto boundary which gave no incentive to other agents to lie about their utility functions, which was that problem's analogue of lying about what you believe is 'fair'.)
If we then look back at the human Responder behaviors, we find that the Proposer's expected gains were:
$Invalid LaTeX $\begin{array}{r|c|c} \text{Offer} & \text{Average subject} & \text{High CRT} \ \hline $5 & $4.8 & $4.95 \ \hline $4 & $4.74 & $5.52 \ \hline $3 & $3.15 & $3.57 \ \hline $2 & $2.16 & $2.64 \ \hline $1 & $1.80 & $2.34 \end{array}: TeX parse error: Misplaced \hline$
This does not support that human Responders are trying to implement a slowly declining curve of Proposer gains. It does suggest that human Responders might be implementing a 'fair' response curve: "Give the Proposer around as much in expectation as they tried to offer me." That is, clipping somewhere around $0.80 (average) or $0.67 (high-CRT) of expected value per $1 decrease in the offer, with unusual forgiveness around the $4-offer mark.
This pattern is more vengeful than the proposed LDT equilibrium. But this degree of vengeance may also make ecological sense in a context where some 'dove' agents will accept sub-$3 offers with high probability. In this case, the equilibrium might call for responding much more harshly to $3 offers in order to disincentivize the Proposer from trying to exploit the chance of encountering a dove. E.g., if you accept $3 offers with 70% probability for an expected Proposer gain of $4.90, but more than 1 in 14 agents are 'dove' agents that accept $3 offers with 90% probability for a Proposer gain of $6.30, then a Proposer should rationally offer $3, testing the possibility that you are a 'dove'.
Arguably, the human subjects in this experiment might still have their rationality critiqued on the grounds that the humans do not really have common knowledge of each other's code. But "human subjects in the Ultimatum Game behave a lot like rational agents might if those agents understood each other's algorithms" is quite a difference from "human subjects in the Ultimatum Game behave irrationally". Perhaps humans simply have good ecological knowledge about the distribution of other agents they might face; or something like an equilibrium among LDT agents emerged evolutionarily and humans now instinctively behave as part of it.
The acceptance rate for $8/$2 splits is much higher if the experimenter is known to have imposed a maximum $2 offer on the Proposer. As Falk et. al. observed, this behavior makes no sense if humans have an innate distaste for unfair outcomes, but in an LDT scenario the answer here changes to "Accept $2." So as an LDT researcher would observe that the human subjects are behaving suggestively like algorithms that know other algorithms are reasoning about them.
Again, one might critique the experimental subjects' rationality on the grounds that they don't in fact have common knowledge of the other agent's code. But the intuitive notion of precommitment warfare suggests that, even if you are facing a robot whose code says to reject all offers below $9, you should offer $9 with at most 55% probability--at least if you believe that robot was designed, rather than it being a natural feature of the environment. Otherwise, a self-modifying agent facing you could self-modify into a simple algorithm that rejects all offers below $9; that is, the corresponding program equilibrium would be unstable. So if humans are behaving like they are part of an equilibrium of LDT agents reasoning about other LDT agents, their uncertainty about exactly which other algorithms they are facing may not imply that humans should 'rationally' accept lowball offers.
The author of the initial version of this page does not know whether a similar equilibrium concept of 'fairness' has been suggested in the rather large literature on the Ultimatum Game. But the author notes that the equilibrium suggested here comes more readily to mind, if you don't think of the rational answer as $1. If anything except accepting $1 is 'irrational', an agent trying to throw a 'usefully irrational' tantrum might as easily demand $9 as $5. LDT results like robust cooperation in the oneshot Prisoner's Dilemma given common knowledge of algorithms are much more suggestive that rational agents in an Ultimatum Game might do something elegant and stable, away from the $9/$1 equilibrium.
LDT equilibria are also arguably a better point of departure from which to view human reasoning, since human behaviors on Ultimatum Game variants are nothing like EDT or CDT equilibria.
The author suggests that LDT analyses of Ultimatum Games--particularly as they regard 'fairness'--might have widespread implications for how we can think economically about non-market bargaining and division-of-gains in the real world.
