We remember how, during the pandemic, companies appeared that cheated on masks and disinfectant fluids. The social and political arrangement was aware of a vague future liability that might or might not come. Then another opportunity to create a monopoly appeared: a national food group was established, whose only achievement was a drastic increase in sugar prices. The combination of politics—specifically, people able to influence legal regulations—and a lack of scruples usually results in restrictions intended to create a quasi-monopoly from which “friends of the establishment” profit.
The situation was different with imports of agricultural produce from Ukraine. There, the restriction was the blockade of Black Sea ports. The country was therefore forced to direct its flow of goods to neighboring countries, including Poland. Everyone remembers what happened next. To this day, the minister of agriculture has not decided to disclose the list of companies that falsified transport documents. Technical grain became consumption grain. What had been dispatched as transit cargo ended up in Polish food wholesalers.
W-MOSZCZYNSKI-2024-12-50Are commercial scams beneficial to the economy?
Unfortunately, Polish economic thought does not concern itself with whether certain phenomena are beneficial to economic development. Breaking the law is always reprehensible, both in Poland and in Western countries. In post-Soviet countries such as Belarus, Russia, Ukraine and Kazakhstan, illegal or fraudulent activity is closely connected with a corrupt system. Scams can be carried out; one merely has to share the proceeds with politicians.
Western countries are different. The law can also be broken there, but one must be fully aware that if the practice is detected, punishment is inevitable. Western culture does not completely condemn companies that try to exploit loopholes and market limitations. A so-called risk premium is always added when calculating such a business. Companies that bear greater operating risk also earn greater profits. We may now ask whether cooperation with a company that operates at the boundary of the law, bears high risk but earns significantly higher profits is worthwhile.
By 2024, the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel had been awarded to 96 laureates. Of these, 60 were citizens of the United States, representing approximately 62.5% of all laureates. Perhaps it is worth looking among them for someone who might try to explain the significance of dishonest companies in the food industry—that is, the sector in which things in Poland have recently become interesting.
Lloyd S. Shapley was an American mathematician and economist known primarily for his work in game theory and mathematical economics. He received the Nobel Prize in Economics in 2012 jointly with Alvin Roth for their contributions to the theory of stable allocations and the practice of market design. He created a method for fairly distributing profits in a cooperative game, based on players’ contributions to different coalitions. It is one of his best-known achievements in game theory. It therefore appears that we have found what we were looking for.
Let us now play with an example from Poland
We have four key players purchasing fruit from the market. Two additional players appear; thanks to political connections, they do something ordinary companies cannot do—they buy fruit from Ukraine at a low price. The fruit is accepted as transit cargo. The documents are then altered and the fruit remains in the country. Each player can shape prices independently. The players can also form coalitions.
The two companies purchasing fruit from Ukraine break the law and bear risk. Because they bear risk, they also earn greater profits. These profits are associated with the risk premium. All six companies may form alliances and contribute additional value.
There is one further condition: if more than three companies unite and begin operating together, they may be fined for monopolistic practices. I do not know whether such a law exists; I added it for amusement. Various restrictions relating to capital structure, legal personality or membership of religious associations have a direct influence on the creation of market distortions. One example is the right of religious associations to purchase agricultural land without having to meet the rigorous conditions imposed on other business entities in Poland.
Mathematical analysis
The formula for the Shapley value
The Shapley value for player i in a game with characteristic function v is defined as:
φᵢ(v) = Σ [ |S|! (|N| − |S| − 1)! / |N|! ]
S ⊆ N {i}
× (v(S ∪ {i}) − v(S))where:
- N—the set of all players (N = {A, B, C, D, E, F}),
- S—any coalition not containing player i,
- |S|—the number of players in coalition S,
- v(S)—the value of coalition S,
- v(S ∪ {i})—the value of coalition S after player i is added,
- |S|!(|N|−|S|−1)! / |N|!—a weight proportional to the number of possible permutations of the coalition.
Everyone can now take offense at me: apparently, one formula in a book causes that book’s sales to fall by half. Following the example of our Nobel laureates, therefore, we could try to explain the importance of dishonest companies through literature. If, however, our laureates are of no use in this field, I propose using formulas offered to us by American Nobel laureates.
Determining the Shapley value is based on carefully calculating each player’s contribution to all possible coalitions, taking into account their effect on total profit. The steps, formulas and detailed calculations are explained below.
The Shapley formula is very simple, and so is its reasoning. We shall now discuss what its individual elements mean.
N: Set of all players
This is the group of all companies participating in the game. In our example: N={A, B, C, D, E, F}. Thus, N denotes all fruit-purchasing companies: A, B, C and D (operating legally), and E and F (operating riskily).
S: Any coalition of players that does NOT include player i
Coalition S is a group of companies that cooperate with one another, but it does not contain the particular company i for which the Shapley value is being calculated. For example, if we calculate the Shapley value for company A, S may be {B, C} or {D, E}, but does not contain A.
|S|: Number of players in coalition S
This is simply the number of companies in a given group S. For example, if S={B, C, D}, then |S|=3: the group contains three companies—B, C and D.
v(S): Value of coalition S
The value v(S) denotes the profit earned by group S when its companies cooperate. For example, if coalition S={B, C} earns a profit of 25, then v(S)=25.
v(S∪{i}): Value of coalition S when player i is added
This means checking the profit when a particular company i joins group S. For example, if S={B, C} earns a profit of 25, but profit increases to 35 when A joins this group, then v(S∪{A})=35.
Marginal contribution
The difference between v(S∪{i}) and v(S) is the “contribution” player i makes to a given coalition:
v(S ∪ {i}) − v(S)For example, if v(S∪{A})=35 and v(S)=25, player A’s contribution is 35−25=10.
Weight
The weight determines how important player i’s contribution is in this particular coalition. It depends on the number of players in S and the number of players in N. The weight formula is:
Weight = |S|! (|N| − |S| − 1)! / |N|!For example, if coalition S contains 2 companies (|S|=2) and there are 6 companies in total (|N|=6), the weight is:
2! (6−2−1)! / 6! = 2!·3! / 6! = 12/720 = 1/60Shapley value
To calculate the Shapley value for player i, we add all marginal contributions from every possible coalition, multiplied by the corresponding weights:
φᵢ(v) = Σ [Weight × (v(S ∪ {i}) − v(S))]
S ⊆ N {i}In other words, we add all of player i’s contributions to different coalitions while accounting for the probability that player i will join a particular group.
Company A, named PUH “Dzbanek”
Suppose PUH “Dzbanek” is one of the honest companies complaining about unfair competition. Instead of fighting, however, “Dzbanek” can begin thinking and attempt to unite with dishonest competitors. Let us see what the calculations would show.
The Shapley value for company A shows how much it contributes to different groups (coalitions) when cooperating with other companies. We calculate how much additional profit A contributes to each possible group and how that contribution affects total profit.
Company A can cooperate with different groups, but first we calculate its contribution when it joins other groups. The possible groups (coalitions) are:
- empty group: S=∅,
- group with one company: S={B}, {C}, …,
- group with two companies: S={B, C}, {B, D}, …,
- group with three companies: S={B, C, D}, …,
- group with four companies: S={B, C, D, E}, … .
For every group, we calculate:
- The value of the coalition without company A: how much the companies in the group earn when A is not with them.
- The value of the coalition with company A: how much the companies earn when A joins them.
- The marginal contribution: the difference between those two values: v(S∪{A})−v(S). For example, if S={B, C} and v(S)=25, while v(S∪{A})=35, company A’s marginal contribution is 35−25=10.
- The weight for each coalition: how many times a given group can appear in different configurations. The weight is calculated from the number of companies in the group.
Calculation examples
Group S=∅ (empty coalition):
- v(S)=0—the empty group generates no profit.
- v(S∪{A})=10—company A alone contributes a profit of 10.
- Marginal contribution: 10−0=10.
- Weight: 0!·5!/6!=120/720=1/6.
- Contribution to the Shapley value: 10·1/6=1.67.
Group S={B}:
- v(S)=10—company B contributes a profit of 10.
- v(S∪{A})=25—companies A and B together earn 25.
- Marginal contribution: 25−10=15.
- Weight: 1!·4!/6!=24/720=1/30.
- Contribution to the Shapley value: 15·1/30=0.5.
Group S={B, C}:
- v(S)=25—companies B and C together earn 25.
- v(S∪{A})=35—adding company A increases profit to 35.
- Marginal contribution: 35−25=10.
- Weight: 2!·3!/6!=12/720=1/60.
- Contribution to the Shapley value: 10·1/60=0.167.
Group S={B, C, D}:
- v(S)=45—the profit of three companies without A.
- v(S∪{A})=55—the profit after company A joins.
- Marginal contribution: 55−45=10.
- Weight: 3!·2!/6!=12/720=1/60.
- Contribution to the Shapley value: 10·1/60=0.167.
Summing the contributions
We add all contributions to the Shapley value:
φA = 1.67 + 0.5 + 0.167 + … = 28.33Summary of the calculation
- Company A contributes additional profit to different groups, as shown by the calculations for each coalition.
- These contributions are weighted to account for how frequently company A appears in different arrangements.
- The total value of the contributions (28.33) shows that company A, PUH “Dzbanek,” is an important but moderately profitable coalition partner.
Is it worthwhile for company A (PUH “Dzbanek”) to enter a coalition with company E (the dishonest company)?
When cooperating with company E (dishonest):
- If A enters a coalition with B and E, the profit is optimal: coalition A, B, E has a value of 65.
- E contributes a premium from cheaper fruit, while risk remains controlled because the number of companies does not exceed 3, so there is no antitrust penalty.
For large coalitions (with more than 3 companies):
- If A cooperates with B, E and F, the antitrust penalty reduces the coalition’s value: coalition A, B, E, F has a value of 95−20=75.
- The increase in value is small compared with the risk associated with the additional companies.
When cooperating only with legal companies:
- Coalition A, B, C has a value of 55.
- Cooperation with legal companies is stable, but profits are lower than when E is added.
Optimum coalition for company A:
Coalition A, B, E is the most profitable:
- it combines the stability of one legal company (B) with the high profits from E;
- it avoids the risk of antitrust penalties because it has fewer than 4 companies;
- it provides a clear increase in value—65 compared with 55 for legal companies alone.
Final conclusion
It is worthwhile for company A, PUH “Dzbanek,” to enter a coalition with E (the “dishonest” company) if it limits coalition membership to a maximum of three companies. Company E then contributes additional profits while the risk remains acceptable. With larger groups, however, the penalty and legal risk outweigh the benefits.
Wojciech Moszczyński
Wojciech Moszczyński—graduate of the Department of Econometrics and Statistics of Nicolaus Copernicus University in Toruń; specialist in econometrics, finance, data science and management accounting. He specializes in optimizing production and logistics processes. He conducts research into the development and application of artificial intelligence. For years, he has been engaged in popularizing machine learning and data science in business environments.
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