Banzhaf Value

The Banzhaf value is one of the most widely known solution concepts of cooperative game theory. It presents a reasonable expectation of the share-out of the global winnings among players and a fair division in normative contexts. It was initially introduced by Lionel Sharples Penrose in 1946 as a power index for voting games and later rediscovered by John Banzhaf in 1965 and James S. Coleman in 1971. Since then, the Banzhaf value has been extended to arbitrary (nonsimple) cooperative games (thus being converted from a power index to game value) by Guillermo Owen, Pradeep Dubey, Lloyd Stowell Shapley, and others. A definition of this value, in absolute and normalized versions, is given here, followed by an illustrative example.

Let N= {1, 2,…, n} be a set of players. A cooperative game v on N is expressed in characteristic function form if a function v (the characteristic function) is given that assigns a value v(S) to every coalition S ⫅ N such that v (Ø) = O.

For each coalition S of N and for each player i belonging to S the “marginal contribution” of such a player to S is the difference between the winning of S and the winning of S without player i—that is, v(S) - v(S\{i}).

The absolute Banzhaf value (β'(v)) assigns to each player i the average of the marginal contributions to all coalitions to which the player belongs:

In this definition, all coalitions are considered equally probable and each player is equally likely to enter any coalition. The absolute Banzhaf value does not satisfy the efficiency property—that is, the sum of the values assigned to all players is not generally equal to the total worth of the grand coalition v(N).

The normalized Banzhaf value (β (v)) assigns to each player i a quota of the total win v(N) proportional to the sum of all h players' marginal contributions to all coalitions:

where K is the quotient between v(N) and the total of marginal contributions of all players; h is the index of players; the first summation in the denominator is over all players h in N

The restriction of the (absolute or normalized) Banzhaf value to simple games is known as the (absolute or normalized) Penrose, Banzhaf, or Penrose-Banzhaf-Coleman measure of voting power.

An Example

Consider the three-person game v({l})= v({2})= v({3})= 0, v({1, 2}) = v({2, 3}) = 1, v({1, 3}) = 2, v({1, 2, 3}) = 3. Let us compute the absolute Banzhaf value of player 1. There are four coalitions containing player 1: {1}, {1,2}, {1, 3} and {1, 2, 3}. The marginal contributions of player 1 to these coalitions are

Thus

Let us compute the absolute Banzhaf value of player 2. There are four coalitions containing player 2: {2}, {1, 2}, {2, 3} and {1, 2, 3}. The marginal contributions of player 2 to these coalitions are 0, 1, 1, 1.

[Page 54]To calculate the normalized Banzhaf values, we divide the absolute Banzhaf values by the coefficient . This gives β1(v) = 15/13, β2(v) = 9/13,β3(v) = 15/13. Note that the sum of these values gives exactly v{1, 2, 3}) = 3.

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