Since Nash's theorem ensures that any n-player finite game will offer at least one rational solution by way of
a Nash equilibrium, it is tempting to conclude that such games offer no further surprises compared to the two-player case. However, consideration of even a simple 3-player example, simple majority, shows that (as for the move from zero-sum to general sum games) the introduction of additional players leads to qualitatively different behaviour.
Simple Majority is formulated in section 21 of Theory of games and economic behaviour1 as follows:
Each player, by a personal move, chooses the number of one of the two other
players. Each one makes his choice uninformed about the choices of the two other players. After
this payments will be made as follows: If two players have chosen each other's numbers we say that
they form a couple. Clearly there will be precisely one couple, or none at all. If there is precisely
one couple, then the two players who belong to it get one half unit each, while the third (excluded)
player correspondingly loses one unit. If there is no couple, then no one gets anything.
Analysed by way of Nash Equilibria, it can be seen that the only irrational situation would be for
no couple to form. Denoting a choice of Player a by Player 1, Player b by Player 2 and Player c by
Player 3 as the triple (a, b, c), this would correspond to a set of strategies (2, 3, 1) or (3, 1, 2) with
a payoff to each of 0. Any other set would create a couple between two players and leave the third
excluded. The six remaining states (2, 1, 2), (2, 1, 1), (3, 1, 1), (3, 3, 1), (2, 3, 2), and (3, 3, 2) are all
Nash equilibria since they contain couples. This is because the excluded player cannot escape their
fate by a unliateral change of strategy, since either strategy fails to match them to a Player who
chose them; whilst a change of strategy by a coupled player either places them in a new couple
(such as Player 1 moving from (3, 1, 1) to (2, 1, 1)) or reduces their payoff from 1/2 to 0 (such as
Player 1 moving from (3, 3, 1) to (2, 3, 1)): Hence no player gains an advantage from unilateral
changes in strategy. Of course, a couple may still fail to emerge since without binding agreements
no player can tell who they should attempt to form a couple with.
As Von Neumann and Morgenstern note, this simple example gives an argument against laissez faire capitalism- despite the absolute, formal fairness of this game due to the symmetry of its rules,
there is no reason why the usage of those rules by the participants, nor the outcomes they receive,
will be fair. Worse, the players are driven to create unsymmetric outcomes since it is rational and
advantageous for them to form couples.
We can view the formation of a couple in the simple majority game as a zero sum payoff by
considering the couple as a single entity. Then the couple gains what the excluded player loses;
in this case, a payoff of 1 unit. The issue then is how that payoff should be divided between the
couple. It was observed that in the above formulation, players are driven to form a couple, but they
are indifferent as to which player they form it with. This was because the distribution of the payoff
was fair within the couple, regardless of which arises. If this condition is relaxed, then players may
gain an incentive to form a particular couple; although which is not immediately obvious.
For instance, suppose the payoff is altered in the event of a couple forming between Player 1 and
Player 2 to 1/2+ε and 1/2-ε respectively. Then Player 1 may appear to favour forming a couple
with Player 2 to one with Player 3. However, if they expect Player 2 to behave rationally, then
they must conclude that Player 2 will opt instead to try and form a couple with Player 3, for a
payoff of 1/2 instead of 1/2 - ε. Hence Player 1 must also choose Player 3, and settle for a payoff
of 1/2 or risk becoming the excluded player and incurring a cost of 1.
In such a scenario, Player 3 is then a favoured partner, and can form a couple with the player of
their choice. Thus it is unreasonable to assume that they will settle for a payoff of 1/2 since they
can safely threaten to form a couple with the other player. In effect, Players 1 or 2 will need to
buy their way into a partnership with Player 3; Player 2 will accept any extra price ε' that is less
than the ε it would cost them to form a partnership with Player 1, for instance.
Thus a range of behaviours can arise which depend upon both the payoff to the couple, and the
distribution within the couple, which could be subject to complex bargaining between the players.
Such scenarios are particularly relevant to economic problems, and to model them more precisely
than with Nash equilibria it is necessary to build upon the notion of couple in the simple majority game to develop a
theory of coalitions and the payoff to individuals within them: this gives rise to cooperative game theory. For such a treatment of simple majority, see coalitional form of a strategic form game.
1Von Neumann and Morgenstern- Theory of Games and Economic Behaviour, Princeton University
Press.
Part of A survey of game theory- see project homenode for details and links to the print version.