Throughout history, people have wondered how to divide resources fairly.
From water shared among travelers lost in the desert to bread placed on the table for a hungry stranger, the dilemma remains the same: how much should each person receive if we want to be truly just?

This problem appears in many cultures. One of the oldest versions can be found in medieval Arabic manuscripts: two travelers had with them five and three loaves of bread. They meet a third traveler and decide that all three will eat from the pooled bread. After the meal, the newcomer, grateful, leaves eight coins for the meal received. How should the money be divided?

In Romanian literature, Ion Creangă revisits the same dilemma in the story “Cinci pâini”, where the simplicity of the situation hides a subtle problem of distributive justice.

In June 2024, we set a better-than-AI level, and to this day no AI model has provided a correct answer to this problem. This fact became the motivation for the present article: to track how long it will take before AI models can solve it correctly.

Below we will explore the logic of fair division, how the dilemma can be solved, and the limitations of the model.

The Problem in E.T. Jaynes Style (Maximum Entropy) Link to heading

5:3

We will present the problem formally, where every piece of information is justified and what is unspecified is judged by the principle of maximum entropy (E.T. Jaynes).

  1. Known information (ex-ante data): (1.1) The first traveler, named A, has 5 liters of water. (1.2) The second traveler, named B, has 3 liters of water. (1.3) The third traveler, named C, has no water, and at the moment of meeting, no one has drunk any water. (1.4) All travelers have the same initial wealth, to remove any pre-existing advantage or disadvantage. (1.5) Before the desert, A and B buy water from the same store at the same price, considering this the optimal strategy to cross the desert.

  2. Unknown information and E.T. Jaynes assumptions: (2.1) There is no additional information about the travelers’ exact preferences or how C decides to behave. (2.2) By the maximum entropy principle, any unknown information must be treated uniformly: we cannot arbitrarily favor anyone in the absence of data.

  3. Situation at the end of the journey: (3.1) All three survive at the limit, consuming water equally, which reflects optimal resource use and the risks taken. (3.2) At the end, C offers M = 8 coins to the two travelers as a reward, without specifying how to split them.

The Classical Solution Link to heading

The intuitive classical solution assumes that rewards are divided proportionally to the net contribution of A and B, without considering the symmetry of the initial situation or the maximum entropy principle.

  1. Volume and proportion calculations:
  • Total water volume:

$$ V = v_A + v_B + v_C = 5 + 3 + 0 = 8 $$

  • Net contribution after equal sharing:

$$ c_A = v_A - \frac{V}{3} = 5 - \frac{8}{3} \approx 2.3333 $$

$$ c_B = v_B - \frac{V}{3} = 3 - \frac{8}{3} \approx 0.3333 $$

  1. Classical division rule:
  • By simplification, the coins received are considered to reflect each person’s net water contribution.
  • Thus, the 8 coins are split proportionally to contributions:

$$ c_A : c_B = \frac{7}{3} : \frac{1}{3} = 7 : 1 $$

Constructive Critique Link to heading

Today, we know that the division is not unique, and applying the classical criterion can create arbitrage opportunities.

Moreover, it should be noted that without B’s input, A would not receive any coins, and could even perish due to lack of water (3.1) if deciding to go alone with C. This aspect cannot be ignored.

If the reward were recorded by invoice, A would legally collect the money, and the split would be decided a priori and transparently. Without this documentation, a judge would consider collecting the money illegal and should not propose an alternative solution.

Introducing money transforms the problem from a Crusoe-style barter economy into a context where any resource, including water, can be monetarily evaluated. This aspect cannot be ignored.

Thus, if the classical solution were legal and the water price, for example, was 1L = 1 coin, arbitrage opportunities arise that can be identified and analyzed.

The Economic Solution Link to heading

To eliminate arbitrage, B—being an actuary—proposes S1 to buy 1 L of water from A at the transfer value before saving the situation and dividing the resources among three. Otherwise, it is considered too risky to enter this arrangement, as it would involve an asymmetric game with a risk of credibility loss.

This move resets the asymmetry between the two, and entry into the game is made under fair conditions, eliminating any arbitrage possibility. Note that at this step, A and B do not know if or what reward they will receive.

This strategy is Pareto efficient: A or B lose nothing from this transfer.

If the transaction is accepted, the initial asymmetry is eliminated, and the 4:4 split is hard to contest, with B net benefiting by 3 coins (4-1), compared to just 1 in the classical solution. Of course, the solution depends on the transfer value of water.

The Physical Solution Link to heading

In physics, symmetries are not just aesthetic properties; they indicate invariances of natural laws. Noether (1918) demonstrated a crucial result: each continuous symmetry of the action of a physical system corresponds to a conservation law — Time translation → energy conservation.

This shows that symmetry precedes law: if you correctly identify the symmetry, the law (and the conserved quantity) follows almost inevitably.

Applying the same idea here, we identify the initial imbalance between A and B and correct the situation by transferring one liter of water from A to B before the evolutionary splitting of the coins.

Thus, a global symmetry is restored, similar to the conservation principle in a closed physical system, which allows the application of a fair and consistent measurement criterion, leading to the same Pareto-optimal solution as the economic one: equal division between A and B, followed by adjustment for the initial water transfer.

Hence, once a global symmetry is established, a reasonable and consistent criterion can be applied, leading to the same solution as the economic approach.

The classical solution does not depend on all initial conditions (transfer value of water) and therefore cannot generate the optimal solution regardless of the system’s initial state. Even if the transfer value of water were zero, the classical solution would still result in the 7:1 split. The same holds if it were worth millions.

System Variation Link to heading

As David Deutsch emphasizes, a good explanation is one that preserves the variation of possibilities and system coherence. In the context of our problem, a solid explanation must adapt to initial conditions but be rigid enough to prevent system exploitation (arbitrage free).

For example, if the transfer value of water were 5 coins per liter, after transaction S1, B would experience a net effect from 1 coin in the classical solution to -1. This variation is expected and reflects how the system responds to changing parameters. Additionally, it shows that S1 does not advantage anyone.

In contrast, the classical solution does not satisfy the variation criterion: it does not provide a robust explanation of results when system parameters change.

Returning to the economic solution, if the classical split were imposed by law, A and B could generate immediate arbitrage after collecting coins from C by proposing an alternative solution S2 that favors them, for example by avoiding taxes or similar mechanisms. Moreover, they could artificially create a player C and, knowing the conditions well, arbitrate even more efficiently.

Conclusions Link to heading

Once a global symmetric framework is established, one can then apply a criterion (economic, physical, or probabilistic) to predict or estimate a future outcome. However, the division is never unique: different criteria can lead to distinct solutions, and arbitrage possibilities persist as long as the division rule is not established before receiving the reward.

We cannot accept sacrificing beneficial solutions for market stability merely for the sake of process simplification.

This approach practically defines the domain of Actuarial Mathematics, especially through credibility theory. test You do not need to master all details—just find the best certified actuary to build the appropriate model!