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Quantitative investing

Shapley values

Shapley values are a concept borrowed from game theory. Regarding the latter, the game constitutes a setting in which players contribute to an overall outcome (payoff). In terms of this, Shapley values are used to distribute the gains or costs fairly across several players cooperating in coalitions. Effectively, they are the average marginal contributions (gains or costs) for all players, when all possible coalitions have been taken into account.


glossary-shapley-values.png

Where

φi is the Shapley value for player i
N is the set of all players and N\{i} is the set of all players excluding player i
n is the number of players in set N: the total number of all players
S is any subset of the set of all players excluding player i
v(S) is the payoff for a coalition consisting of the players in set S

For example, take two individuals (A and B) who would like to Uber home after work. If they do so separately, the cost for A is EUR 10 and for B EUR 15 given varying distances. However, if they share the ride, the fare amounts to EUR 20. With Shapley values, you can calculate how this should be split fairly between the two individuals. In this scenario, the players (A and B) form a coalition (carpool) and they play a game (Uber ride) to obtain a payoff/cost (EUR 20 fare).

In this setting, the Shapley values amount to the players’ average marginal contributions (or costs) across all possible orders in which the players can join the carpool arrangement, i.e., if A hails the ride first or if B is the first to request it. If A books the Uber, her marginal contribution will be EUR 10. Given that B joins the carpool as the second individual, his contribution will also be EUR 10 as that the total cost of the ride is EUR 20. But if B hails the ride, his marginal contribution will be EUR 15 and A will contribute the remaining EUR 5. Therefore, the Shapley values would then amount to EUR 7.5 for A – (10+5)/2 – and EUR 12.5 for B – (10+15)/2.

Shapley values have many applications, including machine learning (ML). In terms of our quant investing platform, we use them in interpreting our ML models. For example, they can help us to determine which variables (features) are the most impactful in our models and rank them in terms of importance. They can also shed light on the outputs generated by our models by deconstructing the marginal contributions of features, thus allowing us to better understand the economic rationale behind our model predictions.


See also

Gradient descent
LASSO regression


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Important information This disclaimer applies to any documents and the verbal or written comments of any person in presentations or webinars on this website and taken together is referred to herein as the “Information”. The services to which the Information relate are NOT FOR RETAIL CLIENTS - The information contained in the Website is solely intended for professional investors, defined as investors which (1) qualify as professional clients within the meaning of the Markets in Financial Instruments Directive (MiFID), (2) have requested to be treated as professional clients within the meaning of the MiFID or (3) are authorized to receive such information under any other applicable laws and must not be relied or acted upon by any other persons. This Information does not constitute an offer to sell, or a solicitation of an offer to buy, any financial product, and may not be relied upon in connection with the purchase or sale of any financial product. You are cautioned against using this Information as the basis for making a decision to purchase any financial product. To the extent that you rely on the Information in connection with any investment decision, you do so at your own risk. The Information does not purport to be complete on any topic addressed. The Information may contain data or analysis prepared by third parties and no representation or warranty about the accuracy of such data or analysis is provided.
In all cases where historical performance is presented, please note that past performance is not a reliable indicator of future results and should not be relied upon as the basis for making an investment decision. Investors may not get back the amount originally invested. Neither Robeco Institutional Asset Management B.V. nor any of its affiliates guarantees the performance or the future returns of any investments. If the currency in which the past performance is displayed differs from the currency of the country in which you reside, then you should be aware that due to exchange rate fluctuations the performance shown may increase or decrease if converted into your local currency. Robeco Institutional Asset Management B.V. (“Robeco”) expressly prohibits any redistribution of the Information without the prior written consent of Robeco. The Information is not intended for distribution to, or use by, any person or entity in any jurisdiction or country where such distribution or use is contrary to law, rule or regulation. Certain information contained in the Information includes calculations or figures that have been prepared internally and have not been audited or verified by a third party. Use of different methods for preparing, calculating or presenting information may lead to different results. Robeco Institutional Asset Management UK Limited (“RIAM UK”) is authorised and regulated by the Financial Conduct Authority. RIAM UK, 30 Fenchurch Street, Part Level 8, London EC3M 3BD (FCA Reference No:1007814). The company is registered in England and Wales under Ref No. 15362605.

In all cases where historical performance is presented, please note that past performance is not a reliable indicator of future results and should not be relied upon as the basis for making an investment decision. Investors may not get back the amount originally invested. Neither Robeco Institutional Asset Management B.V. nor any of its affiliates guarantees the performance or the future returns of any investments. If the currency in which the past performance is displayed differs from the currency of the country in which you reside, then you should be aware that due to exchange rate fluctuations the performance shown may increase or decrease if converted into your local currency. Robeco Institutional Asset Management B.V. (“Robeco”) expressly prohibits any redistribution of the Information without the prior written consent of Robeco. The Information is not intended for distribution to, or use by, any person or entity in any jurisdiction or country where such distribution or use is contrary to law, rule or regulation. Certain information contained in the Information includes calculations or figures that have been prepared internally and have not been audited or verified by a third party. Use of different methods for preparing, calculating or presenting information may lead to different results. Robeco Institutional Asset Management B.V. is authorised as a manager of UCITS and AIFs by the Netherlands Authority for the Financial Markets and subject to limited regulation in the UK by the Financial Conduct Authority. Details about the extent of our regulation by the Financial Conduct Authority are available from us on request.