Abstract
We investigate cases of preference change in the context of cake-cutting problems. In some circumstances, believing that some other player can be credited with a particular preference structure triggers a preference shift by imitation. As a result of this, players may experience regret. However, in typical examples the extent of the change (and the ensuing regret) cannot be anticipated, so players cannot adjust their behavior beforehand. Our goal is to describe the phenomenon, provide a formal model for it, and explore circumstances and allocation procedures that may alleviate some of its negative consequences. In the face of utility shifts we propose a new criterion for fairness, which we dub Ratifiability; in a ratifiable allocation rational players are happy to stick to their choices, in spite of the changes in utilities they may experience. We argue that this embodies a sense of fairness that is not captured by other properties of fair allocation.
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Notes
A few words on our use of this expression. In some cultural backgrounds, where everyday concepts and language are heavily permeated by a psychoanalytic tradition, mostly everybody would recognize himself or herself as neurotic to a certain degree; even if the speaker does not take psychoanalytic theory seriously, it is still in the language—much in the way expressions such as “the sun rises” superseded their pre-Copernican origin. In this line, we would like to stress that by ‘neurosis’ we do not mean to point to a mental illness or to a mental disorder, but to a state of mind in which the person does not know exactly what s/he wants.
A correct assessment of the extent to which dissatisfaction with standard algorithms is due to Preference-Shift Regret requires empirical testing. There might be circumstances that bolster preference changes by imitation, while other circumstances might help prevent this from happening. We hope to address these problems in a future paper. In any case, regardless of how common the phenomenon actually is, we believe there are paradigmatic examples of Preference-Shift Regret that merit theoretical analysis.
Sometimes a distinction is drawn between a piece of cake and a portion of cake (where a piece refers to a connected subset of the cake, and a portion of cake is a collection of pieces). We will not make use of that distinction here; the algorithms we will favor will assume a single cut.
Sometimes assessments of further potential applications of a given procedure are made only subjunctively or even counterfactually—say, with the purpose of evaluating the gain that a player could obtain if the cake were to be reassembled and divided again at \(t_{n+1}\) (which, depending on the circumstances, may or may not be actually feasible). Related to this, we will not make provisions on how players might come to think of re-applying a procedure; this could well be the task of a third party agent. A third party agent may be in charge of making sure the players follow a given procedure in the first place, and, eventually, she may offer them to re-apply it. Actually, we need not cast it as an “offer” at all. Applying, or re-applying a given procedure need not be the result of a decision made by the players; we may think of it as a given feature of the situation they are in.
Strictly speaking, the change is assumed to occur with every behavioral trigger. However, as a matter of fact we will not deal with procedures that allow players to perform more than one action per round, hence we will not need to consider more than one utility shift per round. This can be easily fixed, if needed. In Sect. 5 we offer computer simulations that show how such a \(u^{*}\) can be generated.
Just to clarify: our model is tailored to take care of utility changes by imitation; the way utility shifts are assumed to work helps us circumscribe our object study. Of course other types of shifts are logically possible, but fall outside the scope of the paper. In particular, if players were to change preferences by moving away from the perceived utilities of the other player, they will not feel Preference-Shift Regret in the first place. We come back to this point in further sections.
The definition asks us to consider the expected utility of the future pieces that would be obtained by a re-application that takes place right after the (first) procedure ends—where a re-application of a given procedure may involve once again several rounds.
In Austin’s moving knife procedure, for example, the player who yells “stop!” may have decided to take a fifty-fifty chance to obtain a much valuable piece (assuming the pieces are assigned by tossing a fair coin); cf. Austin (1982).
Exact and Non-manipulable explicit allocation procedures are trivially ratifiable also under the assumption that the cake loses part of its value as time goes by, as is obvious.
An interesting work on sequential cut and choose can be found in Nicolò and Yu (2008), although it is not designed to take care of shifting preferences.
Some of the results we discuss below can also be obtained by assuming that players are extremely risk averse, rather than truthful—with some caveats. We go back to the relation between risk aversion and truthfulness in the next pages.
The last expression is needed for coherence reasons, to make sure that, for every partition of the cake, the sum of the new values of the pieces adds up to the new value of the whole cake.
We do not intend to say that Minimality is a reasonable constraint for every time discount function in every possible context, but that it certainly makes sense in a cake-cutting scenario. Minimal discounts might be just linear, or exponential up to a certain point (where the value becomes zero), among other possibilities.
Utility functions are already temporally indexed, which spares us the need to include the number of rounds elapsed as a second argument of d.
Conservative players do not act out of spite. For example, suppose i’s current gain at round n is 0; i also believes that j’s piece at n has positive value, but that the whole cake will be worthless for both players at \(n+1\). Our conservative principle tells us that, if it is up to i to decide whether to stop or to go on, i will stop at n.
By ‘linear’ we mean that there is a fixed discount \(\delta\) such that, for every round n, the value of the cake C is \(1-\delta (n-1)\). Then the values of the pieces are proportionally adjusted: the utility \(u_{i,n+1}(X_i)\) of piece \(X_i\) in round \(n+1\) would be \((u^{*}_{n,i}(X)/u_{n,i}(C)) (1 - \delta (n-1))\). Given that \(\delta\) is fixed, the value of the cake will be zero at some point.
Note that assuming extreme risk aversion instead of truthfulness is sufficient to prove Proposition 3 (for fully myopic agents with a linear discount); for Proposition 4 (for myopic agents with linear discount) we need common knowledge of extreme risk aversion.
See the discussion at the end of Sect. 3
See the Appendix for a summary of the different procedures mentioned in this paper.
GI does not generalize easily to more than two players, as in such settings it is no longer well defined what a utility change by imitation is. We hope to address generalizations to more than 2 people in future work.
Thanks to an anonymous referee for raising this concern.
Thanks to an anonymous reviewer for pointing out this implication.
Myopic players need not be completely unaware of the fact that they may experience new shifts in the future, but they may choose not to pay attention to that possibility, in the absence of more concrete information.
Isn’t extreme risk aversion always the rational attitude to hold, when information is scarce? (for example, for myopic players who are aware of possible shifts, but have no credences for that?) Not necessarily. If you know nothing (and this includes: if you have no positive probability for your being systematically prone to imitative utility changes) you may as well decide to keep playing now and hope to stop the next time, and hence benefit from the opportunity to make up for what went wrong in this round.
We ran identical simulations with exponential discounts, 0.95 and 0.97, with very similar results. An interesting question is whether there are ratifiable algorithms for which the probability to stop in the first round is positive even when the discount is small or zero (thanks to an anonymous referee for raising this question). To this effect we run simulations for an iterated version of Symmetric Cut and Choose. For one set of simulations we assumed that agents were fully myopic; for a second set of simulations we assumed myopia, but not full myopia. If Player 1 is fully myopic she believes that, in the next round, Player 2 will cut by the same point as he did before. Hence, once Player 1 experiences a utility shift, she can get a larger piece by approaching Player 2’s cutting point; in this scenario, when the discount is zero, agents will always prefer to try another round. The situation changes, however, when agents are just myopic: Player 1 fears that Player 2 may shift his cutting point by approaching Player 1’s previous cut, and hence Player 1 may end with a smaller piece of cake. In this case, even with a zero discount, agents stopped in the first round with probability \(\approx\)0.6. A similar phenomenon applies when the discount is positive. For example, when the discount is 4, the agents will stop with probability larger than 0.95 (compared to \(\approx\)0.2 in the original ICC).
Arguably, in such scenarios utility changes can always be traced back to changes of credences, but this is not our problem here.
This also holds for irrational changes of credences, of course; there is a vast literature that addresses this problem, which is only tangentially related to ours.
We could also explore a modification of the so called undercut procedure described in Brams et al. (2012).
Thanks to an anonymous referee for suggesting us to consider this procedure.
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We want to thank two anonymous referees for very useful comments and suggestions. The paper also benefited from discussions with Juan Dodyk, who made a very careful reading of a previous version of this article. Part of this work was supported by Alexander von Humboldt-Stiftung, Bonn, Germany, through a postdoctoral fellowship to Diego Tajer.
Appendix
Appendix
In Table 1 we compare different procedures mentioned in the paper with respect to the satisfaction of Ratifiability and Non-triviality. We have added to the list a new procedure we call ‘Single Moving Knife’. We assume in all cases that players are fully myopic, conservative, truthful, and hold R-utilities.
‘Standard CC’ stands for ‘Standard Cut and Choose’, and ‘Symmetric CC’ stands for ‘Symmetric Cut and Choose’. Symmetric CC can leave both players unhappy, if they both change their utilities simultaneously. This effect cannot be corrected by Iterated Symmetric CC, and hence it does not satisfy Non-triviality (as we have no guarantees that at least one player will be terminally envy-free).
By ‘Standard Moving Knife’ we mean the procedure by Austin (1982). For truthful agents, it guarantees an exact allocation and hence does not make room for utility shifts in the first place (so Ratifiability is satisfied trivially). By ‘Single Moving Knife’ (‘Single MK’) we refer to the procedure due to Dubins and Spanier (1961): a referee moves a knife over the cake, from left to right. The first agent who shouts ‘stop!’ gets the piece at the left of the cake; the other agent takes the other piece.Footnote 32 Single MK satisfies Non-triviality, and once again it is trivially ratifiable - this time in the sense that, if utility changes by imitation take place at all, they cannot generate regret. To see this, suppose player i learns that player j shouts ‘stop’; hence i learns that j values the two resulting pieces equally. If i now changes her utilities by imitation, i still thinks that the piece at the right is at least as good as the piece at the left (at most, they are worth the same for her now).
‘Crazy Cut and Choose’ (‘Crazy CC’) appeared at the end of Sect. 2, and is neither ratifiable nor non-trivial, whereas the Generalized Null Allocation, in Sect. 4, is trivially Ratifiable (it guarantees that all players obtain zero). Finally, ‘Generalized Iteration’, also in Sect. 4, is a generic procedure that guarantees Ratifiability.
A last comment. As proved in Proposition 5, Ratifiability entails Diachronic Envy-Freeness. The converse, however, only holds under stricter assumptions, and, indeed, there are particular procedures and scenarios for which all players are diachronically envy-free, and yet one or more of the clauses of Ratifiability are not fulfilled. By way of illustration, suppose that players engage in an abnormal version of ICC in which the agent in charge of deciding to stop is the one who chooses (notice that this procedure is no longer a possible instance of Generalized Iteration, as it violates clause (3)). As the chooser cannot experience preference-shift regret, she will always decide to stop (her current piece will always represent to her more than one half of the cake discounted once), so she is both terminally and diachronically envy-free. On the other hand, if the cutter suffers a neurotic change and the temporal discount is large enough, she will be diachronically envy-free (for example, if she thinks the value of the cake reduces dramatically from \(t_{n}\) to \(t_{n+1}\)), but will be terminally envious, in the sense that she will rather have the chooser’s current piece, thus violating clause (2) from Ratifiability. In any case, for the procedures mentioned in the table the two properties stand or fall together.
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Cresto, E., Tajer, D. Fair cake-cutting for imitative agents. Soc Choice Welf 58, 801–833 (2022). https://doi.org/10.1007/s00355-021-01375-2
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DOI: https://doi.org/10.1007/s00355-021-01375-2