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Following Axelrod (1967), I demonstrate this graphically by calculating the payoffs for
mutual opposition, which is always the same regardless of the value of x. I also find the mutual
support payoffs for all nine values of x and plot these points in Figure 3. Assigning the two axes
as scales of payoff values for each lobbyist, note that only three mutual support points, those up
and to the right of the two broken lines, jointly provide more benefit to both lobbyists than the
point representing mutual opposition. These occur only when x is set at 3, 4, and 5 which, of
course, are the same three values in Panel A of Figure 2 in the coalition set. But it is the mutual
support equilibrium point at the upper right kink in Figure 3 that is most important. Because this
is further up and to the right than the others, it represents the highest joint payoff for the CFA
and NTU lobbyists on milk pricing, assuming that all terms in Table 1 remain constant. Because
the value of x is chosen by the lobbyists rather than being a random variable, the best solution in
this scenario is for both lobbyists to jointly support a policy proposal at the legislative median x
= 4, form a coalition, share resources, and jointly lobby to enact this proposal.
---- Insert Figure 3 about here ----
It is important to be aware that this entire demonstration so far has only had a single
varying component, the position of the milk pricing proposal, or x. What if another term was
permitted to vary as well, such as the rate of member benefit loss to the National Taxpayers
Union lobbyist? Since the presentation could rapidly become very complicated, I simply change
the rate of member loss value listed in Table 1 from 5 to 3 so that any deviation away from the
group member ideal is less harmful to the NTU lobbyist now than for the CFA lobbyist (the
NTU lobbyist is now less constrained by group members). The new preference curves are
graphed in Panel B of Figure 2. Without taking the time to present new game matrices, simply
subgame perfect equilibrium outcome will be equivalent to the Nash Solution (see Binmore et al. 1987; Osborne and
Rubinstein 1990).
Convention