How Counteroffers Change the Ultimatum Game

Take it or leave it

Source: gnomeandi/Shutterstock

Go ahead. Make my day. – Harry Callahan, effective, unscrupulous, though fictional San Francisco police detective

The Iranians and Persians are excellent at the art of negotiation. – Donald Trump, former President of the United States

The ultimatum game is an experimental microcosm of negotiation. Proposer P suggests how a small sum of money should be divided and Responder R agrees to the deal or vetoes it. A fair split is typically accepted, whereas splits strongly favoring the proposer are rejected. When that happens, neither P nor R receive anything (Güth et al., 1982; see also Krueger, 2016 and 2020 on this platform). Psychological research focuses on whether, why, and when R might veto a deal and how P might anticipate and avoid this eventuality. The former question tends to turn the game into an issue of moral psychology; the latter question addresses issues of social cognition such as mentalization, theory of mind, and predictions under uncertainty.

After the two steps of proposal and response, the ultimatum game is exhausted. The players go home and the researchers write a paper. This is the game’s beauty and limitation. In the wild, negotiations often go beyond two steps. Let’s consider a game in which the veto power returns to P. Here it is: P offers to split $10. R can accept the proposal or make a counteroffer, which P can then accept or veto.

Suppose P offers an 8:2 split. In the regular game, R is tempted to reject it out of spite, envy, moral outrage, or any combination of these sentiments. Not being able to veto the deal, R can make a counteroffer. This might be a 5:5 split, which had been hoped for in the first place, or it might be 2:8, an equally biased, and now evidently spiteful, counteroffer. A 2:8 counteroffer is psychologically tantamount to a veto. R just lets P draw the consequences (for an alternative interpretation, see the note at the end of this essay). A 5:5 counteroffer is morally superior because it highlights the norm of fairness that R expects both P and R to respect. Vetoing a fair counteroffer reveals P’s selfishness. Being able to foresee all this, P is more likely to offer a fair split in this modified game than in the canonical two-step game. Adding this additional step and allowing both players to make an offer, while leaving the veto power with the first mover, might solve the ultimatum game with a shift towards distributional justice.

In this modified game, P’s veto power is more symbolic than real because rejecting a fair deal is detrimental to both the player’s material and reputational interests (Krueger et al., 2020). Indeed, one could argue that this modified game is moot because even if P offered 6:4, R would likely counter with 5:5 which P would then pretty much have to accept – and therefore be almost certain to offer 5:5 in the first place. To guard against a descent into triviality, consider the possibility that P is allowed to respond to a fair counteroffer by re-asserting the initial offer and thus returning the veto power to R. In this modified modification of the game, we might see the following sequence of events: P offers 8:2 and R counters with 5:5, which P can accept or veto, or insist on the original 8:2 offer. For P to insist on 8:2 is a double dare because it is already clear that R does not like it. Compared with the regular game, P can be more sure now that R will veto 8:2. Therefore, P should not insist on 8:2 and settle for 5:5. Again, even if veto power ultimately rests with R, it seems that even this non-trivial modification of the game, which grants both players an opportunity to make an offer, increases the chances of distributional fairness to prevail.

If my intuitions are correct, the answer to this post’s byline is "yes." You (both of you) will be better off in a counter-ultimatum game because it is more likely that a deal will be reached. Now remember that the canonical design of the game, which does not allow a counteroffer, is the experimenter’s arbitrary creation. Players in the wild can design (or co-design) their own games. Who will stop you from making a counteroffer when presented with an ultimatum?

In the wild, things often happen fast. There is hope that with a little education in game theory, we might realize which game we are in at the time it is being played so that we can generate the best response. Alas, we often realize too late what the game was, particularly if we ended up empty-handed. Then we can promise ourselves to do better next time or rationalize our decision in moralistic terms so that we can live with the material loss.

Note. I had seemingly dismissed the possibility of R countering an 8:2 offer with an equally unfair 2:8 offer. There is, however, a rationale for doing just this. An offer of $2 suggests that P thinks R should be happy to accept this small amount. Indeed, anyone should accept such a small offer because $2 is better than $0. And this inference includes P. R can thus say "If you think I accept $2, I can infer that you too would settle for it. So here I offer $2 to you.” This rationale does not require spite, envy, moral outrage or any other moral emotion. Deductive logic is enough.