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2024年6月19日发(作者：源码集市站)
shapley值公式解释
The Shapley value is designed to address the problem of how
to fairly divide the worth generated by a cooperative effort
among the participants. It seeks to answer the question: "How
much does each player contribute to the overall value, and how
should that value be distributed?" The value can be financial,
political, or any other measure of worth. To calculate the
Shapley value, we look at all possible permutations of the
players and calculate the incremental worth of each player's
participation.
The formula for calculating the Shapley value is as follows:
ϕi=Σs∈Sv(s)-v(s-i)
Where:
- ϕi represents the Shapley value of player i.
- Σs∈S represents the sum over all possible permutations
of the players.
- v(s) represents the worth produced by a particular
permutation s.
- v(s - i) represents the worth produced by all players
except for player i.
The Shapley value takes into account the marginal
contribution of each player within every possible coalition. It
considers the different possible orders of player arrivals and
determines the worth added by each player at each stage. The
formula then takes the average of all these contributions to
calculate the fair value for each player.
v({Alex}) = $50,000
v({Ben}) = $70,000
v({Chris}) = $80,000
v({Dave}) = $60,000
v({Alex, Ben}) = $130,000
v({Alex, Chris}) = $140,000
v({Alex, Dave}) = $120,000
v({Ben, Chris}) = $190,000
v({Ben, Dave}) = $180,000
v({Chris, Dave}) = $200,000
v({Alex, Ben, Chris}) = $250,000
v({Alex, Ben, Dave}) = $230,000
v({Alex, Chris, Dave}) = $260,000
v({Ben, Chris, Dave}) = $310,000
v({Alex, Ben, Chris, Dave}) = $400,000
To calculate the Shapley value for each player, we consider
all possible permutations of the players:
For Alex:
ϕ(Alex) = [v({Alex}) - v(] + [v({Alex, Ben}) - v({Ben})] +
[v({Alex, Chris}) - v({Chris, Ben})] + [v({Alex, Dave}) -
v({Dave, Chris, Ben})]
For Ben:
ϕ(Ben) = [v({Ben}) - v(] + [v({Alex, Ben}) - v({Alex})] +
[v({Ben, Chris}) - v({Chris, Alex})] + [v({Ben, Dave}) - v({Dave,
Chris, Alex})]
Similarly, we can calculate the Shapley value for Chris and
Dave in a similar manner using the formula.
The Shapley value provides a fair way to allocate the total
worth among the players in a cooperative game. It considers the
contribution of each player and gives them credit for their
specific role in creating the overall value. It is widely used
in situations where collaboration and cooperation are necessary,
such as in the distribution of profits among business partners,
the sharing of resources in coalition games, or the allocation
of seats in a parliament.
One of the key advantages of the Shapley value is its
fairness and axiomatic foundation. It satisfies several desired
properties, such as efficiency, linearity, and symmetry. It
ensures that every player receives a fair share based on their
individual contribution without giving undue advantage to any
player. Additionally, the Shapley value is also mathematically
well-defined, making it a robust tool for analyzing cooperative
games.
In conclusion, the Shapley value provides a rigorous method
for dividing the worth generated by a group of players in a
cooperative game. It takes into account the contribution of each
player and provides a fair distribution of the overall value. By
considering all possible permutations of players, it captures
the incremental contribution of each player to the cooperative
effort. The Shapley value has found applications in various
fields and continues to be an important concept in cooperative
game theory.
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