Bargaining in the Fog of Uncertainty

Soit vous me tuez, soit vous me prenez tel que je suis, parce que je serai damné si je change un jour.

[Either kill me or take me as I am, because I'll be damned if I ever change.] ― Marquis de Sade, meditating on the limits of the ultimatum game (back-translated into French with DeepL and in consultation with Achraf Bouzefour)

No one likes an ultimatum — literally “the last” offer — but we often take them, especially if the implied threat in case of refusal is credible. The German economist Werner Güth and his colleagues invented the experimental ultimatum game, in part, because they thought it would be a fun way to show that formal game theory is a poor model of human behavior.

The game fields two players, a proposer P and a responder R. P suggests how to split a sum of money, typically $10. The proposal is an ultimatum because R has to agree to the split, however unfair it might be, else no one receives any money (Güth et al., 1982). Game theory predicts that a rational R accepts any positive sum, however small, because something is better than nothing. However, most Rs reject very low offers, and, expecting this, most Ps offer a sizable share, often 50%.

I have commented on this game several times, and I have found that it is instructive to think about modifications of the standard set-up. In my last post, for example, I explored the implications of allowing offers to be revised if the initial offer is rejected (Krueger, 2025). With that, I was hoping to shed some light on the psychology of second chances more generally.

The ultimatum game is widely accepted as an elegant experimental way to study zero-sum bargaining. One player’s gain is the other player’s loss, and both these outcomes are in plain view. Alas, many, if not most, real-world bargaining situations are murkier. When a car dealer makes a take-it-or-leave-it offer, for example, you may assume that it is to their advantage. The dealer’s self-regarding offer corresponds to an offer of less than 50% on the experimental game.

But just how much below the point of fairness is the dealer’s offer? How could a buyer even know if an offer is fair in the sense that the seller and the buyer benefit equally? The seller typically has more information and experience, and is better positioned than the buyer to know where fairness lies. The seller has an interest in keeping the buyer guessing and eventually mistaking an unfair deal thinking it was a fair one. One heuristic counter-strategy at the buyer’s disposal is to usurp the role of the proposer and submit their own ultimatum (Krueger et al., 2020). The seller’s response will be instructive. If the seller readily accepts the offer, it is probably too generous, leaving the buyer with the winner’s curse. If the seller declines, it is to be hoped that both parties are willing to play a multi-round game so that they may come to an agreement near the actual point of fairness (Bazerman, 2025).

In short, the experimental ultimatum game, though elegant, fails to reflect many real-world bargaining situations because it neglects some critical uncertainties. Another striking feature of the standard game is a difference in the psychology of the Proposer P and the Responder R. If P is interested in maximizing their own share of the money, they have to predict R’s acceptance threshold, and then offer just enough to clear that threshold. To do this, P has to wonder what it is like to be R. Lacking good information on which to run such a mental simulation, P is thrown into a state of uncertainty. By contrast, R only needs to await the offer, contemplate it when it comes, and then consult their own preferences for accepting the offered amount in light of their own fairness needs.

Let us now change the game so that both players face uncertainty with regard to each other. To do this, let us require R to predict the size of P’s offer. If the prediction is reasonably accurate, say within 10%, it’s a deal, otherwise, no player gets paid. R’s challenge is to be a good forecaster instead of just being a potential righteous punisher. Assuming that both players understand this new rule, what are the implications?

R knows that two different types of prediction error are possible. One type of error arises when the prediction is too high. Say, R expects to receive $4 +/- $1 of the standard game amount of $10, but is offered only $2. This offer is now voided by the rules of the game, an outcome akin to what typically happens in the standard game. Another error arises when the prediction is too low. Say, R pessimistically expects $2 +/- $1 and receives an offer of $4. When a surprisingly high offer is voided, R will likely not feel outrage, as in the case of the first error, but shame. R now takes the blame instead of faulting P.

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This modified game, because it asks players to resolve their uncertainties by simulating what the other player will do, leads them into a rabbit hole of infinite regress. When P foresees that a generous offer is at risk of being voided if R fails to expect fairness, P might adjust the offer downward so that it falls near the low bound of R’s prediction zone. In turn, R may expect that a self-interested P will make the offer so low that if falls just inside R’s zone of prediction. To reduce the risk of prediction error, R will in turn lower the prediction and thereby the bounds of the zone of acceptance. This is a move a foresighted P can in turn anticipate.

This dynamic of trying to outpredict each other devolves into a race to the bottom of the range of possible offers. This improves P’s prospects and hurts R’s. One might expect that the modified game, by putting both players in a state of uncertainty, will either erode fairness to the proposer’s advantage or it will result in a greater number of failed games as compared with the standard game. Alternatively, one might suspect that most players cannot think through multiple layers of simulating each other’s simulation (Grüning & Krueger, 2024), and that they wind up settling on deals that are more happily fair and good for the responder.

As long as we await proper empirical tests of these possibilities, we shall remain in a state of uncertainty.