Game Development Reference
In-Depth Information
in every situation and all the players in the game. This is not always
reasonable in, for example, the large auctions played over the internet.
Nash equilibrium presumes that players know what other players are
doing (and are making a best response to it). But how do they gain this
knowledge in a one-shot game, particularly if there are multiple equilibria?
In the following sections, I discuss each of these issues in more detail, and
sketch solution concepts that can deal with them, with pointers to the relevant
literature.
8.2 Robust and resilient equilibrium
Nash equilibrium tolerates deviations by one player. It is perfectly consistent
with Nash equilibrium that two players could do much better by deviating in
a coordinated way. For example, consider a game with n> 1 players where
players must play either 0 or 1. If everyone plays 0, everyone gets a payoff of
1; if exactly two players play 1 and the rest play 0, then the two who play
1 get a payoff of 2, and the rest get 0; otherwise, everyone gets 0. Clearly
everyone playing 0 is a Nash equilibrium, but any pair of players can do
better by deviating and playing 1.
Say that a Nash equilibrium is k -resilient if it tolerates deviations by
coalitions of up to k players. The notion of resilience is an old one in the
game theory literature, going back to Aumann [1959]. Various extensions of
Nash equilibrium have been proposed in the game theory literature to deal
with coalitions [Bernheim et al., 1989, Moreno and Wooders, 1996]. However,
these notions do not deal with players who act in unexpected ways.
There can be many reasons that players act in unexpected ways. One, of
course, is that they are indeed irrational. However, often seemingly irrational
behaviour can be explained by players having unexpected utilities. For
example, in a peer-to-peer network like Kazaa or Gnutella, it would seem
that no rational agent should share files. Whether or not you can get a
file depends only on whether other people share files. Moreover, there are
disincentives for sharing (the possibility of lawsuits, use of bandwidth, etc.).
Nevertheless, people do share files. However, studies of the Gnutella network
have shown that almost 70 percent of users share no files and nearly 50
percent of responses are from the top 1 percent of sharing hosts [Adar and
Huberman, 2000]. Is the behaviour of the sharing hosts irrational? It is if
we assume appropriate utilities. But perhaps sharing hosts get a big kick
out of being the ones that provide everyone else with the music they play.
Is that so irrational? In other cases, seemingly irrational behaviour can be