---
title: "Optimal Lottery"
authors:
  - name: "Charles Dennery"
    affiliation: "London School of Economics"
  - name: "Alexis Direr"
    affiliation: "Université d'Orléans, CNRS, LEO, UMR 7322, and Paris School of Economics"
date: "2014-09-25"
doi: "10.1016/j.jmateco.2014.09.011"
keywords: [decision-making under risk, lottery games, firm behavior, rank-dependent utility, probability weighting, power-law distribution, skewness preference]
jel_codes: [D81, D21]
language: en
type: research-article
journal: "Journal of Mathematical Economics"
---

# Optimal Lottery

**Authors**
- Charles Dennery — London School of Economics
- Alexis Direr — Université d'Orléans, CNRS, LEO (UMR 7322) and Paris School of Economics — *direr@ens.fr*

**DOI**: [10.1016/j.jmateco.2014.09.011](https://doi.org/10.1016/j.jmateco.2014.09.011)

**Keywords**: decision-making under risk, lottery games, firm behavior, rank-dependent utility, probability weighting, power-law distribution.
**JEL codes**: D81, D21.

---

## Abstract

This article proposes an equilibrium approach to lottery markets in which a firm designs an optimal lottery to rank-dependent expected utility (RDU) consumers. We show that a finite number of prizes cannot be optimal, unless implausible utility and probability weighting functions are assumed. We then investigate the conditions under which a probability density function can be optimal. With standard RDU preferences, this implies a discrete probability on the ticket price, and a continuous probability on prizes afterwards. Under some preferences consistent with experimental literature, the optimal lottery follows a power-law distribution, with a plausibly extremely high degree of prize skewness.

---

## 1. Motivation and contributions

The popularity of commercial lotteries with very large jackpots and small winning probabilities reveals a demand for positively skewed lotteries. Skewness preferences are typically rationalised by rank-dependent expected utility (Quiggin 1982), in which agents transform cumulative probabilities. While this demand-side explanation is well established, the supply side has remained largely unaddressed: what form should an optimal lottery take when its operator can freely choose prizes, their probabilities and the number of prizes? Without an answer, applied questions about lottery taxation, dead-weight loss, monopoly regulation, scale economies and prize structure cannot be properly framed (cf. Grote and Matheson 2011).

The paper makes the following contributions.

1. **Negative result on discrete lotteries under RDU.** A profit-maximising lottery with a finite number of prizes requires both the utility function and the probability weighting function to alternate between concavity and convexity each time a new prize is added. In the two-payoff case this reproduces the Friedman–Savage (1948) concave-convex-concave utility shape; for more prizes the implied alternations become empirically implausible. This extends Markowitz (2010), who established the impossibility under expected utility, to the more general RDU framework with an arbitrary discrete number of prizes.

2. **Continuous lotteries are the natural equilibrium under realistic RDU.** Under a concave utility function and an inverse-S shaped weighting function — both consistent with the experimental literature — the optimal lottery has a discrete probability mass on the worst outcome (the ticket price) and a continuous distribution over prizes above it. This rationalises a salient feature of real-world lotteries: tickets are lost with positive probability, and prize values populate a wide continuum.

3. **Power-law prize distribution under empirically motivated functional forms.** With a power utility $u(x)=x^{\sigma}$ and a constant relative sensitivity (CRS) weighting function (Diecidue, Schmidt and Zank 2009; Abdellaoui, l'Haridon and Zank 2010), the optimal cumulative distribution function follows a power law with tail index $\alpha = (1-\sigma)/(1-\gamma)$. The result links probability weighting directly to the demand for power-law distributions discussed in Gabaix (2011).

4. **Calibration on Euromillions data.** Using 378 consecutive drawings (February 2004 – May 2011), the paper estimates the tail index of the empirical prize distribution and compares it with the optimality condition derived under realistic preference parameters.

5. **Connection to risk-sharing literature.** Unlike Chateauneuf, Dana and Tallon (2000), Dana and Carlier (2008), or Bernard, He, Yan and Zhou (2013), consumption risk does not preexist in this environment. The lottery operator creates risk via a randomisation device to satisfy the risk-taking demand of RDU agents.

---

## 2. Setting and preferences

**Lottery.** A discrete lottery is a list of $n$ payoffs $(x_i)_{i=1,\dots,n}$ and $n+1$ cumulative probabilities $(\pi_i)_{i=0,\dots,n}$ with $\pi_0=0$ and $\pi_n=1$. Payoffs lie in an interval $I$ that may be unbounded; the smallest payoffs are negative to ensure a positive expected profit (commercial lotteries embed a ticket price in the lowest outcomes).

**Firm.** Risk-neutral, with profit
$$\Pi = -\sum_{i=1}^{n}(\pi_i-\pi_{i-1})x_i.$$

**Consumer.** RDU decision maker with strictly increasing, $C^{1}$ utility $u$ on $I$ and strictly increasing, $C^{1}$ weighting function $g:[0,1]\to[0,1]$ with $g(0)=0$, $g(1)=1$. Lottery value:
$$U = \sum_{i=1}^{n}\bigl(g(\pi_i)-g(\pi_{i-1})\bigr)\,u(x_i).$$

**Optimisation problem.** Following standard practice in Pareto-optimal contracting, the paper maximises consumer utility under a minimum-profit constraint $\Pi \ge B$, with monotonicity constraints on $(x_i)$ and $(\pi_i)$. This dual problem is equivalent to maximising firm profit under a participation constraint.

A two-stage maximisation is required: (i) for fixed $n$, choose payoffs and probabilities optimally; (ii) optimise over $n$ itself. The number of prizes is endogenous because adding a prize with $x_{i+1}=x_i$ or $\pi_i=\pi_{i-1}$ replicates a smaller lottery — additional prizes can never worsen the outcome but only sometimes improve it.

---

## 3. Discrete lotteries (Section 2)

### 3.1 Optimality conditions (Proposition 1)

Necessary first-order and second-order conditions for an interior selection of $(x_i,\pi_i)$ are:
$$\frac{g(\pi_i)-g(\pi_{i-1})}{\pi_i-\pi_{i-1}}\,u'(x_i)=\lambda, \qquad \frac{u(x_{i+1})-u(x_i)}{x_{i+1}-x_i}\,g'(\pi_i)=\lambda,$$
$$u''(x_i)\le 0,\qquad g''(\pi_i)\ge 0.$$

The first equation says that prizes are scaled up where the chord ratio $[g(\pi_i)-g(\pi_{i-1})]/[\pi_i-\pi_{i-1}]$ exceeds one, i.e. where the operator can exploit probability overweighting. The second condition is the dual statement on probabilities, absent from Quiggin (1991), which assumed equally probable tickets.

### 3.2 Exclusion conditions (Proposition 2)

For an *optimal* lottery (and not merely an $n$-optimal one), no additional prize can profitably be inserted between two existing prizes. This yields two geometric conditions.

- If a candidate prize $x_i$ between $x_{i-1}$ and $x_{i+1}$ is optimally merged with $x_{i+1}$, the chord ratios of $g$ on either side of $\pi_i$ must satisfy
$$\frac{g(\pi_i)-g(\pi_{i-1})}{\pi_i-\pi_{i-1}} \;\ge\; \frac{g(\pi_{i+1})-g(\pi_i)}{\pi_{i+1}-\pi_i},$$
which means $g$ must be **concave** between two selected probabilities while being locally convex around each of them.

- Similarly, if a candidate $x_i\in(x_{i-1},x_{i+1})$ is excluded by setting $\pi_i=\pi_{i-1}$, the chord ratios of $u$ must satisfy
$$\frac{u(x_i)-u(x_{i-1})}{x_i-x_{i-1}} \;\le\; \frac{u(x_{i+1})-u(x_i)}{x_{i+1}-x_i},$$
which means $u$ must be **convex** between two selected payoffs while being locally concave around each of them.

> **Authors' critical reading.** The shape of the utility function required for a finite-prize optimum is concave-convex-concave between *every pair* of consecutive selected payoffs, and the weighting function must be convex-concave-convex between *every pair* of consecutive selected probabilities. The number of curvature alternations grows linearly with the number of prizes. While this is conceivable with two payoffs (the Friedman–Savage 1948 case), it becomes implausible for richer prize structures. The negative conclusion is that under standard RDU preferences (concave $u$, inverse-S shaped $g$), a discrete-prize lottery is generically dominated by adding more prizes.

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## 4. Continuous lotteries (Section 3)

### 4.1 Continuous problem

The setting is extended to all cumulative distributions on a closed interval $[a,b]$ (which may be arbitrarily large):
$$\max_{F}\; U(F) = \int_a^b f(x)\,g'(F(x))\,u(x)\,dx \qquad \text{s.t.} \qquad -\int_a^b f(x)\,x\,dx = B.$$

Appendix B establishes four general properties of the continuous problem:

| Property | Statement |
|---|---|
| 1 — Existence | Optimal lottery exists when $[a,b]$ is closed and bounded (uses Helly's selection theorem). |
| 2 — Uniqueness | Holds when $u$ is strictly concave or $g$ strictly convex. |
| 3 — Constructibility | Optimal lottery is the pointwise limit of $n$-optimal discrete lotteries. |
| 4 — Continuity | The CDF is strictly increasing wherever $u$ is strictly concave and $g$ strictly convex; jumps wherever $g$ is strictly concave; remains constant wherever $u$ is strictly convex. |

### 4.2 Realistic RDU preferences (Proposition 3)

Under a strictly concave $u$ and an inverse-S shaped $g$ with inflection point $\delta$ (Camerer and Ho 1994; Bleichrodt and Pinto 2000; Abdellaoui 2000; Etchart-Vincent 2004; Tversky and Kahneman 1992; Wu and Gonzalez 1996):

- **(i)** The smallest payoff $x_0\ge a$ has a discrete probability mass $\pi_0\ge\delta$, with slackness $(x_0-a)(\pi_0-\delta)=0$.
- **(ii)** The maximum payoff $x_1\le b$ has a discrete probability $1-\pi_1\ge 0$, with slackness $(b-x_1)(1-\pi_1)=0$.
- **(iii)** The distribution is continuous on $(x_0,x_1)$ and characterised by the first-order condition
$$g'(F(x))\,u'(x) = \lambda.$$

Two empirically meaningful predictions follow. First, the minimal prize concentrates a probability mass — it absorbs the entire concave part of the weighting function — which mirrors the fact that real lottery tickets are lost with positive probability. Second, prizes above the ticket price are continuously distributed. Differentiating the FOC gives the density:
$$f(x) = -\frac{u''(x)}{u'(x)}\cdot\frac{g'(F(x))}{g''(F(x))}.$$
The prize distribution is more spread out the lower the curvature of $u$ or the higher the convexity of $g$.

### 4.3 Power-law distribution (Proposition 4)

Specify $u(x)=x^{\sigma}$ with $0<\sigma<1$ on the gain domain, and a CRS weighting function
$$g(\pi) = \begin{cases}\delta^{1-\gamma}\pi^{\gamma} & 0\le\pi\le\delta \\ 1-(1-\delta)^{1-\gamma}(1-\pi)^{\gamma} & \delta<\pi\le 1\end{cases}$$
with $0<\gamma<1$. The CRS family has constant relative sensitivity $1-\gamma$ on the right tail; $\gamma$ measures responsiveness to changes in cumulative probabilities close to $1$ (smaller $\gamma$ ⇒ more sensitive), while $\delta$ governs elevation/optimism. Under these specifications the optimal CDF is a power law with tail index
$$\alpha = \frac{1-\sigma}{1-\gamma}.$$

The two preference channels work in opposite directions: a more concave utility (smaller $\sigma$) raises the cost of spreading prizes and thins the upper tail (higher $\alpha$); a smaller $\gamma$ amplifies overweighting of close-to-zero probabilities and thickens the upper tail (lower $\alpha$).

---

## 5. Empirical calibration on Euromillions

**Source.** French operator data (www.francaise-des-jeux.fr).
**Sample.** 378 consecutive drawings, February 2004 – May 2011.
**Volumes.** 3,616 prize payouts; 611,269,599 winners; 12 winning ranks.
**Prize range.** From €5 to €129,818,429.

The pari-mutuel design — total prizes equal to a fixed share of ticket sales, divided across 12 winning ranks whose populations vary stochastically — generates a near-continuous empirical prize support, despite the discrete game design.

A linear regression of $-\log(1-F(x))$ on $\log(x)$ yields:

$$-\log(1-F(x)) = 1.378 + 1.066\,\log(x), \qquad R^2 = 0.94.$$

The estimated tail index is $\hat\alpha = 1.066$.

### 5.1 Consistency with experimental parameter values

Empirical estimates from the experimental literature give $\gamma\approx 0.5$ (Abdellaoui, l'Haridon and Zank 2010) and $\sigma\in[0.3,0.9]$ (Tversky and Kahneman 1992; Wu and Gonzalez 1996; Camerer and Ho 1994; Fennema and van Assen 1998; Abdellaoui 2000; Abdellaoui, Bleichrodt and Paraschiv 2007). Plugging into Prop. 4 gives a predicted tail index in the range $[0.2, 1.4]$. The Euromillions estimate $\hat\alpha=1.066$ lies inside this interval. As benchmarks, Atkinson and Piketty (2007) report tail indices between 1.5 and 3 for income, and Kleiber and Kotz (2003) about 1.5 for wealth — lottery prizes are markedly more skewed than the wealth or income distribution.

> **Authors' critical reading.** The calibration is described as a "back-of-the-envelope" exercise. It shows consistency of orders of magnitude rather than a structural test, and rests on the assumption of a time-invariant prize distribution, which is violated when jackpots roll over. Removing rollovers would mostly affect extreme-gain observations.

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## 6. Conclusion

Lotteries are characterised as Pareto equilibria between a risk-neutral firm and RDU consumers. The main results are:

1. Discrete-prize lotteries are incompatible with standard RDU preferences (concave $u$ and inverse-S $g$). Optimality requires implausible alternations of curvature in both functions.
2. Continuous lotteries are the natural equilibrium: a probability mass on the lowest payoff (the ticket price) and a continuous distribution above.
3. With power utility and a CRS weighting function, the prize distribution follows a power law with explicit tail index $\alpha=(1-\sigma)/(1-\gamma)$.
4. Calibration on Euromillions yields $\hat\alpha\approx 1.07$, consistent with experimentally measured preference parameters.

### Limitations and research extensions

The authors flag two qualifications. First, although discrete lotteries are not optimal, the welfare loss relative to a continuous optimum may be small — only a calibration would settle this quantitative question. Second, real lotteries are necessarily discrete because of practical and cognitive constraints not captured by the model: physical implementation of the random device, complexity costs, and possible distrust of opaque many-prize structures. Behavioural channels beyond RDU (e.g. ambiguity aversion towards complex random processes) might explain why lotteries with very many prizes do not dominate fewer-prize formats in practice. These open the way for further work bridging the gap between observed prize structures and theoretical predictions.

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## Acknowledgments

The authors acknowledge useful comments from Nathalie Etchart-Vincent, Michèle Cohen, Christian Gollier and Paul Schilp, and feedback from participants at Université Panthéon-Sorbonne, Paris School of Economics, the Far East and South Asia Meeting of the Econometric Society (2009), the World Risk and Insurance Economics Congress (2010) and the European Meeting of the Econometric Society (2011).

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## Main references

Abdellaoui, M. (2000). Parameter-free elicitation of utility and probability weighting functions. *Management Science* 46, 1497–1512.

Abdellaoui, M., Bleichrodt, H., Paraschiv, C. (2007). Loss Aversion under Prospect Theory: A Parameter-Free Measurement. *Management Science* 53(10), 1659–1674.

Abdellaoui, M., l'Haridon, O., Zank, H. (2010). Separating Curvature and Elevation: A Parametric Weighting Function. *Journal of Risk and Uncertainty* 41(1), 39–65.

Astebro, T., Mata, J., Santos-Pinto, L. (2011). Preference for Skew in Lotteries: Evidence from the Laboratory. Working paper.

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Barberis, N. (2013). The Psychology of Tail Events: Progress and Challenges. *American Economic Review Papers and Proceedings* 103, 611–616.

Bernard, C., He, X., Yan, J.-Y., Zhou, X.-Y. (forthcoming). Optimal Insurance under Rank Dependent Utility. *Mathematical Finance*.

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Diecidue, E., Schmidt, U., Zank, H. (2009). Parametric Weighting Functions. *Journal of Economic Theory* 144(3), 1102–1118.

Friedman, M., Savage, L. J. (1948). The Utility Analysis of Choices Involving Risk. *Journal of Political Economy* 56(4), 279–304.

Gabaix, X. (2011). Power Laws in Economics and Finance. *Annual Review of Economics* 3, 255–293.

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Quiggin, J. (1982). A Theory of Anticipated Utility. *Journal of Economic Behavior and Organization* 3, 225–243.

Quiggin, J. (1991). On the Optimal Design of Lotteries. *Economica* 58, 1–16.

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

Wu, G., Gonzalez, R. (1996). Curvature of the Probability Weighting Function. *Management Science* 42, 1676–1690.

*The full reference list appears in the PDF.*