This document provides an introduction to n-person game theory, including definitions of characteristic functions, imputations, and solution concepts such as the core and Shapley value. The core is the set of undominated imputations, where an imputation is dominated if another coalition can do better. The Shapley value provides a more equitable solution than the core by considering each player's marginal contributions. Examples calculating the core and Shapley value are provided for an drug invention game.

1. GAME THEORY
CHAPTER
THREE
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2. INTRODUCTION TO N-PERSON GAME
THEORY
In many competitive situations, there are more
than two competitors. With this in mind, we now
turn our attention to games with three or more
players. Let N {1, 2, . . . , n} be the set of players.
Any game with n players is an n-person game.
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3. CONT’D
For our purposes, an n-person game is
specified by the game’s characteristic function.
For each subset S of N, the characteristic
function v of a game gives the amount v(S) that
the members of S can be sure of receiving if
they act together and form a coalition. Thus,
v(S) can be determined by calculating the
amount that members of S can get without any
help from players who are not in S.
The characteristics function must satisfy the
superadditivity
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4. CONT’D
Here are many solution concepts for n-person
games. A solution concept should indicate the
reward that each player will receive. More formally,
let x = {x1, x2, . . . , xn} be a vector such that
player i receives a reward xi. We call such a vector a
reward vector. A reward vector x = (x1, x2, . . . ,
xn) is not a reasonable candidate for a solution
unless x satisfies
If x satisfies both the individual and group
rationality, we say that x is an imputation
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5. EXAMPLE: THE DRUG GAME
Joe Willie has invented a new drug. Joe cannot
manufacture the drug himself, but he can sell the drug’s
formula to Company 2 or Company 3. The lucky
company will split a $1 million profit with Joe Willie.
Find the characteristic function for this game.
Solution
Letting Joe Willie be player 1, Company 2 be player 2,
and Company 3 be player 3, we find the characteristic
function for this game to be:
v(v({ }) = v({1}) = v({2}) = v({3}) = v({2, 3}) = 0
v({1, 2}) = v({1, 3}) = v({1, 2, 3}) = $1,000,000
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Empty
coalition
Grand
coalition

6. EXERCISE
Player 1 owns a piece of land and values the land at
$10,000. Player 2 is a subdivider who can develop
the land and increase its worth to $20,000. Player 3
is a subdivider who can develop the land and
increase its worth to $30,000. There are no other
prospective buyers. Find the characteristic function
for this game.
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7. THE CORE OF AN N PERSON GAME
An important solution concept for an n-person
game is the core. Before defining this, we must
define the concept of domination. Given an
imputation x = (x1, x2, . . . , xn), we say that the
imputation y = (y1, y2, . . . , yn) dominates x
through a coalition S (written y > sx) if
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8. CONT’D
Thus, if y > sx, then x should not be considered a
possible solution to the game, because the players
in S can object to the rewards given by x and
enforce their objection by banding together and
thereby receiving the rewards given by y [because
members of S can surely receive an amount equal
to v(S)].
The founders of game theory, John von Neumann
and Oskar Morgenstern, argued that a reasonable
solution concept for an n-person game was the set
of all undominated imputations.
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9. EXAMPLE
Consider a three-person game with the following
characteristic
function:
v({v({ }) = v({1}) = v({2}) = v({3}) = 0,
v({1, 2}) = 0.1, v({1, 3}) = 0.2, v({2, 3}) = 0.2, v({1,
2, 3}) = 1
Let x = (0.05, 0.90, 0.05) and y = (0.10, 0.80, 0.10).
Show that y > {1,3}x
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10. CONT’D
First, note that both x and y are imputations. Next,
observe that with the imputation y, players 1 and 3
both receive more than they receive with x. Also, y
gives the players in {1, 3} a total of 0.10 + 0.10 =
0.20. Because 0.20 does not exceed v({1, 3}) =
0.20, it is reasonable to assume that players 1 and
3 can band together and receive a total reward of
0.20. Thus, players 1 and 3 will never allow the
rewards given by x to occur.
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11. CONT’D
Determine the core of an n-person game.
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Theorem 1 states that an imputation x is in the
core (that x is undominated) if and only if for
every coalition S, the total of the rewards received
by the players in S (according to x) is at least as
large as v(S).

12. FIND THE CORE OF THE DRUG
GAME
For this game, x = (x1, x2, x3) will be an imputation if and only if
x1 ≥ 0
(1)
x2 ≥ 0
(2)
x3 ≥ 0
(3)
x1+ x2 + x3 = $1,000,000
(4)
Theorem 1 shows that x = (x1, x2, x3) will be in the core if and
only if x1, x2, and x3 satisfy (1)–(4) and the following inequalities:
x1 + x2 ≥ $ 1,000,000
(5)
x1 + x3 ≥ $ 1,000,000
(6)
x2 + x3 ≥ $ 0
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13. CONT’D
To determine the core, note that if x = (x1, x2, x3) is in
the core, then x1, x2, and x3 must satisfy the inequality
generated by adding together inequalities (5)–(8).
Adding (5)–(8) yields 2(x1 + x2 + x3) ≥ $2,000,000, or
x1 + x2 + x3 ≥ $1,000,000 (9)
By (4), x1 + x2 + x3 = $1,000,000. Thus, (5)–(7) must
all be binding.†
Simultaneously solving (5)–(7) as equalities yields x1 =
$1,000,000, x2 = $0, x3 = $0. A quick check shows
that ($1,000,000, $0, $0) does satisfy (1)–(8). In
summary, the core of this game is
the imputation ($1,000,000, $0, $0). Thus, the core
emphasizes the importance of Player 1
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14. EXERCISE
Player 1 owns a piece of land and values the land at
$10,000. Player 2 is a subdivider who can develop
the land and increase its worth to $20,000. Player 3
is a subdivider who can develop the land and
increase its worth to $30,000. There are no other
prospective buyers. Find the characteristic function
for this game.
Let x = ($19,000, $1,000, $10,000) and y =
($19,800, $100, $10,100).
Find the core of the land development game?
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15. THE SHAPLEY VALUE †
Now we discuss an alternative solution concept for
n-person games, the Shapley value, which generally
gives more equitable solutions than the core. ‡
The core gives all the rewards to the game’s most
important players, however, the shapely value gives
a more equitable solution.
For any characteristic function, Lloyd Shapley
showed there is a unique reward vector x = (x1,
x2, . . . , xn) satisfying the following axioms:
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16. AXIOMS
Axiom 1: Relabeling of players interchanges the players’
rewards. Suppose the Shapley value for a three-person game is
x = (10, 15, 20). If we interchange the roles of player 1 and
player 3 [for example, if originally v({1}) = 10 and v({3}) = 15,
we would make v({1}) = 15 and v({3}) = 10], then the Shapley
value for the new game would be x = (20, 15, 10).
Axiom 2: This is simply group rationality
Axiom 3: If v(S - {i}) = v(S) holds for all coalitions S, then the
Shapley value has xi = 0. If player I add no value to any
coalition, then player I receive a reward of zero from the
Shapley value
Axiom 4: Let x be the Shapley value vector for game v, and let y
be the Shapley value vector for the game . Then the Shapley
value vector for the game (v + ) is the vector x + y.
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17. THEOREM 2
Given any n-person game with the characteristic
function v, there is a unique reward vector x = (x1,
x2, . . . , xn) satisfying Axioms 1–4. The reward of
the ith player (xi) is given by
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where │S│ is the number of players in S, and for
n ≥ 1, n! n(n -1) … 2(1) (0! = 1).
1
In (1)
2

18. FIND THE SHAPLEY VALUE FOR
THE DRUG GAME
To compute x1, the reward that player 1 should
receive, we list all coalitions S for which player 1 is
not a member. For each of these coalitions, we
compute v(S ∪ {i}) - v(S) and p3(S) (see the following
Tables). Because player 1 adds (on average)
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Table 1 Table 2

19. CONT’D
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the Shapley value concept recommends that
player 1 receive a reward of
To compute the Shapley value for player 2, we
require the information in the table in the
previous slide. Thus, the Shapley value
recommends a reward of
for player 2. The Shapley value must allocate a
total of v({1, 2, 3}) $1,000,000 to the players, so
the Shapley value will recommend that player 3
receive $1,000,000 – x – x2 =

20. CONT’D
1. Recall that the core of this game assigned
$1,000,000 to player 1 and no money to players
2 and 3. Thus, the Shapley value treats players 2
and 3 more fairly than the core. In general, the
Shapley value provides more equitable solutions
than the core.
2. For a game with few players, it may be easier to
compute each player’s Shapley value by using the
fact that player i should receive the expected
amount that she adds to the coalition present
when she arrives. For Example 11, this method
yields the computations in Table 2. Each of the
six orderings of the arrivals of the players is
equally likely, so we find that the Shapley value
to each player is as follows:
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21. CONT’D
3. The Shapley value can be used as a measure of
the power of individual members of a political or
business organization. For example, the UN
Security Council consists of five permanent
members (who have veto power over any resolution)
and ten t members. For a resolution to pass the
Security Council, it must receive at least nine votes,
including the votes of all permanent members.
Assigning a value of 1 to all coalitions that can pass
a resolution and a value of 0 to all coalitions that
cannot pass a resolution defines a characteristic
function. For this characteristic function, it can be
shown that the Shapley value of each permanent
member is 0.1963, and of each nonpermanent
member is 0.001865, giving 5(0.1963) +
10(0.001865) = 1. Thus, the Shapley value
indicates that 5(0.1963) = 98.15% of the power in
the Security Council resides with the permanent
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22. CONT’D
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23. END
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