Nessah and Tian prove that an SNE exists if the following conditions are satisfied: • The strategy space of each player is
compact and
convex; • The payoff function of each player is
concave and continuous; • The
coalition consistency property: there exists a weight-vector-tuple
w, assigning a weight-vector
wS to each possible coalition
S, such that for each strategy-profile
x, there exists a strategy-profile
z in which
zS maximizes the weighted (by wS) social welfare to members of
S, given
x−
S. • Note that if
x is itself an SNE, then
z can be taken to be equal to
x. If
x is not an SNE, the condition requires that one can move to a different strategy-profile which is a social-welfare-best-response for all coalitions simultaneously. For example, consider a game with two players, with strategy spaces [1/3, 2] and [3/4, 2], which are clearly compact and convex. The utility functions are: • u1(x) = - x12 + x2 + 1 • u2(x) = x1 - x22 + 1 which are continuous and convex. It remains to check coalition consistency. For every strategy-tuple x, we check the weighted-best-response of each coalition: • For the coalition {1}, we need to find, for every x2, maxy1 (-y12 + x2 + 1); it is clear that the maximum is attained at the smallest point of the strategy space, which is y1=1/3. • For the coalition {2}, we similarly see that for every x1, the maximum payoff is attained at the smallest point, y2=3/4. • For the coalition {1,2}, with weights w1,w2, we need to find maxy1,y2 (w1*(-y12 + y2 + 1)+w2*(y1 - y22 + 1)). Using the
derivative test, we can find out that the maximum point is y1=w2/(2*w1) and y2=w1/(2*w2). By taking w1=0.6,w2=0.4 we get y1=1/3 and y2=3/4. So, with w1=0.6,w2=0.4 the point (1/3,3/4) is a consistent social-welfare-best-response for all coalitions simultaneously. Therefore, an SNE exists, at the same point (1/3,3/4). Here is an example in which the coalition consistency fails, and indeed there is no SNE.There are two players, with strategy space [0,1]. Their utility functions are: • u1(x) = -x1 + 2*x2; • u2(x) = 2*x1 - x2. There is a unique Nash equilibrium at (0,0), with payoff vector (0,0). However, it is not SNE as the coalition {1,2} can deviate to (1,1), with payoff vector (1,1). Indeed, coalition consistency is violated at
x=(0,0): for the coalition {1,2}, for any weight-vector
wS, the social-welfare-best-response is either on the line (1,0)--(1,1) or on the line (0,1)--(1,1); but any such point is not a best-response for the player playing 1. Nessah and Tian also present a necessary and sufficient condition for SNE existence, along with an algorithm that finds an SNE if and only if it exists. == Properties ==