Optimal design of lottery with cumulative prospect theory

Mitsuaki Obara; Shunta Akiyama; Yasushi Kawase

arxiv: 2209.00822 · v2 · pith:IOMKEAOEnew · submitted 2022-09-02 · 💻 cs.GT · econ.TH

Optimal design of lottery with cumulative prospect theory

Shunta Akiyama , Mitsuaki Obara , Yasushi Kawase This is my paper

Pith reviewed 2026-05-24 10:52 UTC · model grok-4.3

classification 💻 cs.GT econ.TH

keywords cumulative prospect theorylottery designseller profit maximizationnon-convex optimizationlinear-time algorithmthree-level optimizationticket price constraint

0 comments

The pith

A linear-time algorithm finds the profit-maximizing lottery when buyers evaluate outcomes according to cumulative prospect theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses how a seller should set prize amounts and probabilities to maximize revenue when buyers decide according to cumulative prospect theory, a model that overweight extreme outcomes and treats gains and losses asymmetrically. The central technical obstacle is that the resulting objective is non-convex, which the authors handle by recasting the design task as a three-level optimization problem whose solutions can be fully characterized. From that characterization they derive a linear-time procedure that returns the optimal lottery and an extension that incorporates a ticket-price constraint. A sympathetic reader would care because the procedure supplies the first explicit method for constructing CPT-optimal lotteries with arbitrarily many outcomes, moving beyond the two-outcome restriction of earlier work.

Core claim

The authors show that the seller's profit-maximization problem under cumulative prospect theory can be reformulated as an equivalent three-level optimization problem; optimal solutions of this reformulation are characterized by a small number of candidate lotteries that can be enumerated in linear time, yielding the first polynomial-time algorithm for designing CPT lotteries with more than two outcomes.

What carries the argument

The three-level optimization reformulation of the CPT objective that converts the original non-convex program into a form whose optima can be exhaustively checked by a linear scan.

If this is right

Sellers obtain the revenue-maximizing prize vector and probability vector for any prescribed number of outcomes in linear time.
The same linear-time procedure extends directly to the variant in which the ticket price itself is a decision variable subject to an upper bound.
Previous two-outcome CPT lottery models are recovered as special cases of the new characterization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Real lotteries could be stress-tested by feeding their published prize tables into the algorithm and comparing predicted versus observed sales.
If empirical CPT parameters vary across buyer populations, the algorithm supplies a fast way to recompute the optimal design for each population.
The three-level structure may be reusable for other non-convex behavioral objectives that admit similar nested reformulations.

Load-bearing premise

Buyers evaluate every lottery exactly according to the cumulative prospect theory value and weighting functions with fixed, known parameters, and the three-level reformulation introduces no extraneous solutions.

What would settle it

An explicit lottery instance in which the three-level algorithm returns a design whose CPT value is strictly lower than that of another feasible lottery with the same number of outcomes.

Figures

Figures reproduced from arXiv: 2209.00822 by Mitsuaki Obara, Shunta Akiyama, Yasushi Kawase.

read the original abstract

Lotteries are a prevalent form of gambling between a seller and buyers. Designing a lottery requires a model of how buyers make decisions when confronted with uncertain outcomes. Cumulative prospect theory (CPT) is a descriptive model that captures people's propensity to overestimate extreme events and their different attitudes toward gains and losses. In this study, we design a lottery that maximizes the seller's profit when the buyers' decision-making adheres to the CPT framework. The main difficulty is the nonconvexity of the CPT framework, which we overcome by reformulating the problem as a three-level optimization problem and characterizing its optimal solution. Based on the analysis, we propose a linear-time algorithm that computes the optimal lottery. Furthermore, we present an efficient algorithm applicable to a broader setting with a ticket price constraint. This is the first study to employ the CPT framework in designing an optimal lottery with more than two outcomes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a linear-time algorithm for optimal lottery design under CPT for more than two outcomes via a three-level reformulation, but the exact mapping back to the original objective needs verification.

read the letter

The one thing to know is that this paper gives a linear-time algorithm for optimal lottery design under cumulative prospect theory for more than two outcomes by recasting the non-convex problem as three-level optimization. It is the first to handle that setting under CPT rather than expected utility or two-outcome cases. They also supply a variant with a ticket price constraint. The CPT parameters are taken as fixed inputs from the behavioral literature, which is the right move and keeps the work free of circularity. The reformulation and solution characterization are the technical steps that let them claim efficiency. That extension and the algorithmic result are the concrete contributions. The soft spot is the three-level reformulation. The algorithm and optimality claim rest on it preserving the original CPT objective and feasible set exactly, with no added or dropped solutions. The abstract states that the reformulation is done and the solution is characterized, but without the equations or proof details the mapping cannot be checked here. If the equivalence holds, the linear-time result is useful; if it does not, the procedure solves something else. The stress-test note correctly flags this as load-bearing. This paper is for algorithmic mechanism designers who want to bring descriptive models like CPT into lottery or mechanism settings. A reader focused on efficient computation under non-expected utility would get value from the reformulation technique and the running-time claims. It is coherent on its own terms and engages the relevant literature, so it deserves peer review. I would send it to referees with a request to check the reformulation step in detail.

Referee Report

2 major / 1 minor

Summary. The paper claims to maximize a seller's profit in designing a lottery when buyers evaluate outcomes according to cumulative prospect theory (CPT). The central technical contribution is a reformulation of the resulting non-convex optimization problem into an exact three-level program whose optimal solution is characterized; this characterization yields a linear-time algorithm for the unrestricted case and an efficient algorithm when a ticket-price constraint is added. The work asserts it is the first to apply CPT to lotteries with more than two outcomes.

Significance. If the three-level reformulation is shown to be equivalent to the original CPT objective and the linear-time algorithm is correct, the result would be significant: it supplies the first computationally tractable method for multi-outcome CPT lottery design and demonstrates that a non-convex behavioral objective can be reduced to a form admitting an efficient exact algorithm.

major comments (2)

[Abstract / three-level reformulation claim] The abstract states that the non-convex CPT problem is overcome by reformulating it as a three-level optimization problem whose solutions coincide with the original. No derivation, proof sketch, or explicit mapping is visible that confirms the reformulation neither introduces extraneous solutions nor drops feasible lotteries while preserving the CPT value function and weighting functions over ranked outcomes; this equivalence is load-bearing for both the optimality characterization and the linear-time algorithm.
[Abstract / linear-time algorithm claim] The linear-time algorithm is asserted to compute the optimal lottery on the basis of the three-level characterization. Without a visible verification that the reformulation is exact, it is impossible to confirm that the algorithm returns a lottery that is optimal under the original CPT objective rather than under a modified objective.

minor comments (1)

[Abstract] The abstract refers to 'characterizing its optimal solution' but does not indicate whether the characterization is closed-form, structural, or algorithmic; a brief statement of the form of the characterization would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below and are prepared to make revisions to improve the clarity of the presentation.

read point-by-point responses

Referee: [Abstract / three-level reformulation claim] The abstract states that the non-convex CPT problem is overcome by reformulating it as a three-level optimization problem whose solutions coincide with the original. No derivation, proof sketch, or explicit mapping is visible that confirms the reformulation neither introduces extraneous solutions nor drops feasible lotteries while preserving the CPT value function and weighting functions over ranked outcomes; this equivalence is load-bearing for both the optimality characterization and the linear-time algorithm.

Authors: The equivalence is rigorously established in the body of the paper (Section 3), where we derive the three-level program from the original CPT objective and prove that their optimal solutions coincide, including the preservation of the value and weighting functions. The abstract is intended as a high-level summary and does not contain technical derivations, which is standard. To address the concern, we will include a short proof sketch in the introduction of the revised manuscript. revision: yes
Referee: [Abstract / linear-time algorithm claim] The linear-time algorithm is asserted to compute the optimal lottery on the basis of the three-level characterization. Without a visible verification that the reformulation is exact, it is impossible to confirm that the algorithm returns a lottery that is optimal under the original CPT objective rather than under a modified objective.

Authors: The linear-time algorithm is proven correct based on the equivalence shown in Section 3 and the optimality characterization in Section 4. The algorithm solves the three-level program, which we prove equivalent, thus guaranteeing optimality for the original CPT objective. We will add a remark in the algorithm description explicitly linking back to the equivalence proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reformulation treated as independent mathematical step

full rationale

The paper's derivation begins from the standard CPT value function and weighting functions (treated as exogenous inputs with fixed parameters drawn from the behavioral literature) and reformulates the non-convex seller-profit maximization as a three-level optimization problem whose solution is then characterized to yield a linear-time algorithm. No equation or step reduces the claimed optimal lottery or algorithm output to a parameter fitted inside the paper, a self-defined quantity, or a self-citation chain. The three-level reformulation is presented as an analytical device to overcome non-convexity rather than a renaming or tautological restatement of the original objective. CPT parameters remain external, and the algorithm's correctness claim rests on the (unverified here) equivalence of the reformulation rather than on any definitional loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that buyers follow a fixed CPT model whose parameters are taken as given; no new entities are introduced and the optimization reformulation is presented as a technical device rather than an additional axiom.

axioms (1)

domain assumption Buyers evaluate lotteries exactly according to cumulative prospect theory with its standard value and weighting functions.
Stated in the abstract as the modeling premise for buyer behavior.

pith-pipeline@v0.9.0 · 5677 in / 1239 out tokens · 25856 ms · 2026-05-24T10:52:15.672719+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The most widely used value function for CPT is V(w) := w^α (w≥0), -λ(-w)^β (w<0) … with α,β,λ … set for the function to be steeper for loss than gain
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We overcome this difficulty by reformulating the problem as a three-level optimization problem … propose a linear-time algorithm

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

Arbiv and Y

T. Arbiv and Y . Aumann. Fair and truthful giveaway lotteries. In Proceedings of the AAAI Conference on Artiﬁcial Intelligence, volume 22, 2022

work page 2022
[2]

H. Aziz, R. de Haan, and B. Rastegari. Pareto optimal allocation under uncertain preferences. In Proceedings of IJCAI, page 77–83, 2017

work page 2017
[3]

N. C. Barberis. Thirty years of prospect theory in economics: a review and assessment. Journal of Economic Perspectives, 27(1):173–96, 2013

work page 2013
[4]

Bernard, X

C. Bernard, X. He, J.-A. Yan, and X. Zhou. Optimal insurance design under rank-dependent expected utility. Mathematical ﬁnance, 25(1):154–186, 2015

work page 2015
[5]

Boyd and L

S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004

work page 2004
[6]

Dennery and A

C. Dennery and A. Direr. Optimal lottery. Journal of Mathematical Economics, 55(1):15–23, 2014

work page 2014
[7]

Fennema and P

H. Fennema and P. Wakker. Original and cumulative prospect theory: a discussion of empirical differences. Journal of Behavioral Decision Making, 10(1):53–64, 1997

work page 1997
[8]

P. C. Fishburn. Utility theory for decision making. Wiley, New York, 1970

work page 1970
[9]

Friedman and L

M. Friedman and L. J. Savage. The utility analysis of choices involving risk. Journal of political Economy, 56 (4):279–304, 1948

work page 1948
[10]

Ghossoub

M. Ghossoub. Optimal insurance under rank-dependent expected utility. Insurance: Mathematics and Eco- nomics, 87:51–66, 2019

work page 2019
[11]

Kahneman and A

D. Kahneman and A. Tversky. Prospect theory: an analysis of decision under risk.Econometrica, 47(2):263–291, 1979

work page 1979
[12]

Kawase and H

Y . Kawase and H. Sumita. On the max-min fair stochastic allocation of indivisible goods. In Proceedings of the AAAI Conference on Artiﬁcial Intelligence, pages 2070–2078, 2020

work page 2070
[13]

A. Maeda. Optimal lottery design for public ﬁnancing. The Economic Journal, 118(532):1698–1718, 2008

work page 2008
[14]

R. B. Myerson. Optimal auction design. Mathematics of operations research, 6(1):58–73, 1981

work page 1981
[15]

J. Quiggin. A theory of anticipated utility. Journal of Economic Behavior and Organization, 3(4):323–343, 1982

work page 1982
[16]

J. Quiggin. On the optimal design of lotteries. Economica, 58(229):1–16, 1991

work page 1991
[17]

M. O. Rieger, M. Wang, and T. Hens. Prospect Theory around the World. Technical Report 2011/19, Norwegian School of Economics, Department of Business and Management Science, 2011

work page 2011
[18]

M. O. Rieger, M. Wang, and T. Hens. Estimating cumulative prospect theory parameters from an international survey. Theory and Decision, 82(4):567–596, 2017

work page 2017
[19]

Roughgarden

T. Roughgarden. Twenty lectures on algorithmic game theory. Cambridge University Press, 2016

work page 2016
[20]

C. Starmer. Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk. Journal of economic literature, 38(2):332–382, 2000. 23

work page 2000
[21]

K. C. J. Sung, S. C. P. Yam, S.-P. Yung, and J. H. Zhou. Behavioral optimal insurance. Insurance: Mathematics and Economics, 49(3):418–428, 2011

work page 2011
[22]

Tversky and D

A. Tversky and D. Kahneman. Advances in prospect theory: cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4):297–323, 1992

work page 1992
[23]

von Neumann and O

J. von Neumann and O. Morgenstern. Theory of games and economic behavior . Princeton University Press, 1944

work page 1944
[24]

Z. Xu, X. Zhou, and S. Zhuang. Optimal insurance under rank-dependent utility and incentive compatibility. Mathematical Finance, 29(2):659–692, 2019. 24

work page 2019

[1] [1]

Arbiv and Y

T. Arbiv and Y . Aumann. Fair and truthful giveaway lotteries. In Proceedings of the AAAI Conference on Artiﬁcial Intelligence, volume 22, 2022

work page 2022

[2] [2]

H. Aziz, R. de Haan, and B. Rastegari. Pareto optimal allocation under uncertain preferences. In Proceedings of IJCAI, page 77–83, 2017

work page 2017

[3] [3]

N. C. Barberis. Thirty years of prospect theory in economics: a review and assessment. Journal of Economic Perspectives, 27(1):173–96, 2013

work page 2013

[4] [4]

Bernard, X

C. Bernard, X. He, J.-A. Yan, and X. Zhou. Optimal insurance design under rank-dependent expected utility. Mathematical ﬁnance, 25(1):154–186, 2015

work page 2015

[5] [5]

Boyd and L

S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004

work page 2004

[6] [6]

Dennery and A

C. Dennery and A. Direr. Optimal lottery. Journal of Mathematical Economics, 55(1):15–23, 2014

work page 2014

[7] [7]

Fennema and P

H. Fennema and P. Wakker. Original and cumulative prospect theory: a discussion of empirical differences. Journal of Behavioral Decision Making, 10(1):53–64, 1997

work page 1997

[8] [8]

P. C. Fishburn. Utility theory for decision making. Wiley, New York, 1970

work page 1970

[9] [9]

Friedman and L

M. Friedman and L. J. Savage. The utility analysis of choices involving risk. Journal of political Economy, 56 (4):279–304, 1948

work page 1948

[10] [10]

Ghossoub

M. Ghossoub. Optimal insurance under rank-dependent expected utility. Insurance: Mathematics and Eco- nomics, 87:51–66, 2019

work page 2019

[11] [11]

Kahneman and A

D. Kahneman and A. Tversky. Prospect theory: an analysis of decision under risk.Econometrica, 47(2):263–291, 1979

work page 1979

[12] [12]

Kawase and H

Y . Kawase and H. Sumita. On the max-min fair stochastic allocation of indivisible goods. In Proceedings of the AAAI Conference on Artiﬁcial Intelligence, pages 2070–2078, 2020

work page 2070

[13] [13]

A. Maeda. Optimal lottery design for public ﬁnancing. The Economic Journal, 118(532):1698–1718, 2008

work page 2008

[14] [14]

R. B. Myerson. Optimal auction design. Mathematics of operations research, 6(1):58–73, 1981

work page 1981

[15] [15]

J. Quiggin. A theory of anticipated utility. Journal of Economic Behavior and Organization, 3(4):323–343, 1982

work page 1982

[16] [16]

J. Quiggin. On the optimal design of lotteries. Economica, 58(229):1–16, 1991

work page 1991

[17] [17]

M. O. Rieger, M. Wang, and T. Hens. Prospect Theory around the World. Technical Report 2011/19, Norwegian School of Economics, Department of Business and Management Science, 2011

work page 2011

[18] [18]

M. O. Rieger, M. Wang, and T. Hens. Estimating cumulative prospect theory parameters from an international survey. Theory and Decision, 82(4):567–596, 2017

work page 2017

[19] [19]

Roughgarden

T. Roughgarden. Twenty lectures on algorithmic game theory. Cambridge University Press, 2016

work page 2016

[20] [20]

C. Starmer. Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk. Journal of economic literature, 38(2):332–382, 2000. 23

work page 2000

[21] [21]

K. C. J. Sung, S. C. P. Yam, S.-P. Yung, and J. H. Zhou. Behavioral optimal insurance. Insurance: Mathematics and Economics, 49(3):418–428, 2011

work page 2011

[22] [22]

Tversky and D

A. Tversky and D. Kahneman. Advances in prospect theory: cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4):297–323, 1992

work page 1992

[23] [23]

von Neumann and O

J. von Neumann and O. Morgenstern. Theory of games and economic behavior . Princeton University Press, 1944

work page 1944

[24] [24]

Z. Xu, X. Zhou, and S. Zhuang. Optimal insurance under rank-dependent utility and incentive compatibility. Mathematical Finance, 29(2):659–692, 2019. 24

work page 2019