Part II Selected Topics
7.4 Coordination, Intelligence, and Nash Equilibrium
rational players to choose (B), it helps to consider what happens if they make any other choice. Making his or her choice independently, Player 1 can only pick one of the two options. From the point of view of Player 1, picking (A) means he or she could end up with a nice lunch if the other player stays in, but they could end up with no lunch if the other player goes out. In contrast, picking (B) means they would get the best lunch if the other stays in, and a passable lunch if the other goes out. So, whichever choice the other player makes, choosing (B) improves Player 1’s lunch.
This example has the same features as The Prisoners’ Dilemma, a classic example in game theory. More background about this game can be found in the article Prisoner’s Dilemma.
7.3.2 The Cost of Lacking Communication and Trust Can Be Unbounded
To convince ourselves of the soundness of the above analysis, it is important to realize that each player must make his or her decision independently. This does not mean that this is what any two people in this situationshould do, rather, it is clarifying how the formal notion of games that we are studying works. We said that we are studying games where each player is trying to maximize utility, and that, for this particular game, the utilities are as shown in the table. We didnotsay anything about players’ ability to communicate or their ability to trust each other; as a result, we have to exclude the possibility of the players coordinating, because the ability to communicate and trust are strong assumptions that we cannot make without changing our original problem statement. In fact, a profound lesson that can be drawn from this example is that the costs of lacking communication and/or trust can be unbounded:
we can replace 4 and 5 in this example by any pair of arbitrarily large values, and as long as the first is less than the second, rationality and self-interest forces both players to choose (B). Lacking communication and trust can be arbitrarily costly for everyone involved.
120 7 Game Theory We also saw how strict dominance can be used to determine that the rational behavior of two players in such a game has to be (BB). At the same time, we also saw that (BB) may not be the highestpossiblepayoff for both players, but it is the highest payoff that they canguarantee independently.
The power of strict dominance lies in its usefulness in narrowing down the set of possible rational strategy pairs to a smaller set. However, it will not always be possible to find strictly dominant strategies (or more specifically, strictlydominated strategies to exclude). It is therefore useful to consider how to interpret games where there are no strictly dominated strategies, and where we have more than one possible rational outcome.
7.4.1 The Coordination Pattern
Consider a two-player game where each player can choose between two strategies:
going to a movie (A) or going to a play (B). Both players only care about being together. Let us call this the Coordination Pattern. The following table represents this pattern:
1\ 2 A B A ++ - - B - - ++
It is clear that in this case there is no strictly dominant strategy: For each player, (A) is better if, and only if, the other player chooses (A), and the same holds for (B).
We have two cases where there is a win-win choice, (AA) and (BB), but achieving either depends critically oncoordination.
When we considered our Small-Auction vs. Fridge Lunch example, we noted that communication and mutual trust would have been needed to arrive at a better outcome than that provided by the dominant strategy. Here, there is no dominant strategy at all. The absence of a dominant strategy can be viewed as the absence of a reward to always unilaterally select one strategy versus the other. In such cases, communication is key. However, trust is no longer necessary: the utilities put both players in a situation where (a) it is in their interestonlyto communicate their intent truthfully and (b) once they have shared their intent, the other player is only motivated to act in a manner that is optimal for both of them.
7.4.2 Nash Equilibrium
Note that this type of reasoning reflectsintelligenceon the part of the players, in the sense that it takes into account that they are aware of the other player’s utilities and
decision-making process. The observation that we can predict the outcomes of games more precisely when we take into account not only each player’s rationality but also their ability to reason about the other player’s decision-making process is attributed to John Nash. It is his name that is acknowledged in the term “Nash Equilibrium,”
which refers to the set of plays (strategy combinations) out of which no player has an incentive to depart unilaterally. In the example above, the set { (AA), (BB) } is the Nash Equilibrium for this game. The game motivates both players to only be in one of these plays. And once they are in one of them, they would only be motivated to move to another onein coordinationwith the other players.
7.4.3 Determining the Nash Equilibrium
With one additional condition, the Nash Equilibrium for a game pattern is simply the set of all strategy combinations with (++) utilities. The extra condition is that each player should only have a plus (+) option as the maximum utility for any one of his strategies. This is the case for both the Lunch and Coordination Patterns. Thus, in the Independence Pattern, the Nash Equilibrium is the set { (BB) }.
Example 7.6: An Asymmetric Four-Strategy GameTo check our understanding of this method of computing the Nash Equilibrium set, we will consider a game with four strategies and asymmetric utilities:
1\ 2 A B C D A 7 1 2 4 4 8 6 4 B 1 3 3 7 5 6 6 2 C 3 2 4 4 7 5 8 3 D 9 7 2 8 1 9 5 3
When we mark the highest utility in each choice for the first player, we get the following table:
1\ 2 A B C D A 7 1 2 4 4 8 6 4 B 1 3 3 7 5 6 6 2 C 3 2 + 4 + 5 + 3 D + 7 2 8 1 9 5 3
When we mark the highest utility in each choice for the second player, we get the following table:
122 7 Game Theory 1\ 2 A B C D
A 7 1 2 4 4 + 6 4 B 1 3 3 + 5 6 6 2 C 3 2 4 4 7 + 8 3 1\ 2 A B C D D 9 7 2 8 1 + 5 3
Combining all the marks into one table we get the following, where the cells that have utility now marked (++) form the Nash Equilibrium set:
1\ 2 A B C D A 7 1 2 4 4 + 6 4 B 1 3 3 + 5 6 6 2 C 3 2 + 4 + + + 3 D + 7 2 8 1 + 5 3
As this table shows, the Nash Equilibrium set is { (CC) }. Thus, if both players reason rationally, taking into account the other player’s utilities as options, the first player would choose (C) and the second would play (C). These are the choices that each player can make independentlyandsecure the maximum possible payoff, given the utilities for the different choices for both players.
7.4.4 Eliminating Strictly Dominated Strategies Preserves Nash Equilibria
In games where there are a large number of possible strategies, it is useful to remove from consideration (or eliminate) strategies that a rational player would never choose.
Strict dominance gives us just the right tool for doing so, as any strictly dominated strategy can be safely eliminated in this manner. What is more, eliminating one choice for one play can reveal other dominated strategies for the other player (since they are also intelligent, and can determine for themselves that the first player would never play that strategy).
This technique is synergistic with the notion of Nash Equilibria: eliminating strictly dominated choices does not remove any elements of the Nash Equilibrium of a game.
Exercise 7.1Remove strictly dominated strategies from the game presented in this last example. Repeat this process until there are no more strictly dominated strategies. Draw the table for the reduced game. Once you have done so, determine the Nash Equilibrium for the reduced game.