The Shapley Value has been around for seven decades now. It is intuitive, tractable, and has many desirable properties. To my surprise, however, its actual applications in the real world are not readily available (as defined by a simple Google search). Most of the "applications" of the Shapley Value I could find are theoretical/hypothetical in nature. For example, the Investopedia entry lists the pricing of airport runway usage and marketing analytics as two applications without reference to actual real world cases.

My question is: Are there any real world applications of the Shapley Value in pricing transactions and/or allocating profits/costs among cooperating parties?

More specifically, I'm looking for answers in any of the following forms:

  • "As documented in [news/professional/academic] Publication X, Company A used the Shapley Value (or its variant) to price its transaction with Company B and Company C."
  • "As documented in [news/professional/academic] Publication X, Company A used the Shapley Value (or its variant) to allocate its [R&D/marketing/overhead] costs among its subsidiaries B, C and D."
  • "The empirical study by Author et al. (20xx) in Publication X found that the profit/cost sharing practice within a [firm/industry/multinational enterprise] is consistent with the prediction of some model based on the Shapley Value."
  • "The empirical study by Author et al. (20xx) in Publication X uses [a structural estimation/an empirical strategy] that is based on the Shapley Value."

Please do provide links/references to the sources. Thank you.

Note 1. While I'm primarily interested in the application of the Shapley Value in a business context, pointers to its use in other domains such as politics would also be appreciated.

Note 2. I found a paper by Littlechild and Thompson (1977) where the authors applied the Shapley Value in calculating the prices of airport runway usage based on actual data from the Birmingham Airport. It is unclear, however, whether those prices were subsequently adopted/implemented by the airport authorities.

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    $\begingroup$ There are many empirical papers in politics that use the Shapley value to measure power. $\endgroup$ – Michael Greinecker Mar 10 at 20:28
  • $\begingroup$ Thanks @Michael. While I'm primarily interested in the application of Shapley Value in a business context, pointers to its use in politics would also be appreciated. $\endgroup$ – Herr K. Mar 10 at 20:59
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    $\begingroup$ If you have access to it, Eyal Winter's chapter in volume 3 of the "Handbook of Game Theory and Economic Applications" by Aumann and Hart discusses several such papers. $\endgroup$ – Michael Greinecker Mar 10 at 21:10
  • $\begingroup$ @MichaelGreinecker: That's great! I'll take a look. Thank you. $\endgroup$ – Herr K. Mar 10 at 21:40
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    $\begingroup$ Shapley values, in the form of SHAP extension, are very actively used in explaining Machine Learning/AI models. Very active area of research. $\endgroup$ – Sergey Bushmanov Mar 16 at 13:07

Allegedly, Barclays uses (or once used) Shapley allocation. Quoting Mauro Cesa:

The basic objective of every bank is to find an optimal business strategy that maximises return on capital (ROC). To this end, banks will allocate more capital to desks that generate the highest ROC, while those with lower ROC receive a smaller share of available capital. This is an intuitive and seemingly sensible solution.

But the authors of this month’s paper, Reduced-form capital optimisation, argue that ranking business units by ROC might not result in an optimal allocation of capital. The approach ignores the correlation between businesses and will deliver the optimal allocation only if the correlation is zero – an assumption for which there is little evidence in the real world.

The problem is further complicated by the Basel III capital rules, which are primarily based on two ratios: risk-weighted assets (RWAs) and leverage balance sheet (LBS). The minimum capital requirement for each legal entity under the same parent bank is the greater of the two.

To optimise its ROC, a legal entity must hold RWA and LBS capital in equal measure. The selection of RWA or LBS, as directed by the greater-of-the-two rule, introduces non-linearities that are difficult to deal with in an optimisation context.

“This is a huge, unsolved problem for banks,” says Yadong Li, managing director of quantitative analytics at Barclays, and one of the authors of the paper. “The bank as a whole ultimately needs to deploy the resources to different business units. This is a key part of senior management’s job.”

Dimitri Offengenden, a Tel Aviv-based managing director of quantitative strategy at Barclays, joined the bank in 2017 specifically to work on the capital allocation problem. He consulted with Li, who has worked on related issues in the past.

“I suggested we look at Shapley allocation, which is a natural way to solve the greater-of-two problem” says Li.

Shapley allocation is well known in game theory and has been widely applied in economics and business decision making. It attributes a value to the contribution of each agent in a system where agents co-operate and share the costs and gains of their activity. In essence, it measures the marginal contribution of each agent by observing the difference the agent’s presence or absence makes to the activity. Based on that, a portion of the shareable pie is allocated to each agent. In the case of a bank, the agents are individual business units and the pie is the overall capital stock.

Offengenden, Li and Jan Burgy, a quant strategist at Barclays, realised that regressing the overall allocation of capital on the selection of RWAs and LBS obtained from the application of the Shapley method resulted in an almost perfect approximation.

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    $\begingroup$ Nice! Thanks for the answer! $\endgroup$ – Herr K. Mar 11 at 2:53

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