The following example illustrates how to transform a strategic form game into coalitional form, so will probably be of little use or interest unless you have read those writeups.
Individual payoffs in the simple majority game with symmetric distribution can be described by the following table, where the rows denote the options for Player 1 and the columns the options for Players 2 and 3 (as an ordered pair):
| (1,1) (1,2) (3,1) (3,2)
---+-----------------------------------------------------
2 |(1/2,1/2,-1) (1/2,1/2,-1) (0,0,0) (-1,1/2,1/2)
3 |(1/2,-1,1/2) (0,0,0) (1/2,-1,1/2) (-1,1/2,1/2)
The value v({1, 2, 3}) is determined by free choice of any of the 8 strategy combinations, but for
any such choice the sum of the payoffs is 0. As always, v(∅) = 0. To determine v({1}) (and by
symmetry v({2}) and v({3})) we can consider Players 2 and 3 as acting as a single entity against
Player 1; the return to their coalition being the sum of the individual returns, which they are
motivated to drive as high as possible. It should be clear then that they will form a couple to
ensure a return of 1, forcing a payoff of -1 onto Player 1, ensuring that v({1}) = -1. However, it
is enlightening to see precisely why this occurs.
By treating Players 2 and 3 as a single player with strategy set A,B,C,D the above table reduces
to
| A B C D
---+--------------------------------------------
2 | (1/2,-1/2) (1/2,-1/2) (0,0) (-1,1)
3 | (1/2,-1/2) (0,0) (1/2,-1/2) (-1,1)
This is a
bimatrix for a 2 player strategic form game; moreover it is the bimatrix of the
zero-sum
game given by
1/2 1/2 0 -1
1/2 0 1/2 -1
Further, column 4 dominates all other columns, so the strategic form reduces to that column- giving
the game a value of -1 for Player 1 (and hence of 1 for the coalition of Players 2 and 3).
So the simple majority game is described in coalitional form by
- v(∅) = v(X) = 0
- v({1}) = v({2}) = v({3}) = -1
- v({1, 2}) = v({1, 3}) = v({2, 3}) = 1
Superadditivity holds, since v({i}) + v({j, k}) = -1 + 1 = 0 ≤ 0 = v({i, j, k}) for any
permutation
i, j, k of the players. Further, the game is zero-sum; but it is
essential, since
v({1}) + v({2}) + v({3}) = -3 < v(X)
Hence the corollary given in the
coalitional form writeup does not generalise to any zero-sum game; it is specific to the two-player case.
The construction of the characteristic function used in the preceding example can be used, after
suitable generalisation, for any strategic form game. Given a coalition S ∈ΒΈ P(X), we consider a
two-player zero-sum game between two teams S and X\S. The strategy sets for each team consists
of the cartesian product of the strategy sets of the individual members of each team. For instance,
in the above example Players 2 and 3 had strategy sets {1, 3} and {1, 2} respectively giving rise to the
combined strategy set {A,B,C,D} = {(1, 1), (1, 2), (3, 1), (3, 2)}. The payoff to the coalition for
any given combination of strategies is then determined by the sum of the payoffs to its members
from those strategies. v(S) is then determined by the value of the game, which (due to the Minimax
theorem) always exists.
Part of A survey of game theory- see project homenode for details and links to the print version.