The Nash Equilibrium
The prisoners' dilemma is far more applicable to a wide range of economic situations and can
thus be analysed from the perspective of a game to determine a strategy that maximises
payoffs. The complexity of this situation is increased by the extent to which there is cooperation
or non-cooperation between the players. Let us go back to the Nash example of the girls in the
bar, which we highlighted earlier. In this example, all four males managed to improve their
position compared to the situation where they all chased the blonde. What the outcome rested
on, though, was the cooperation between them all.
The outcome of the game in this instance depends on the extent to which the cooperation
between the players can be enforced. I trust you implicitly that you will go after one of the
brunettes and you also trust me implicitly, even though we know that we would both prefer the
blonde. What happens, however, if I renege on the agreement and secretly go after the blonde
and win her affection? It may be that the enforceable action in this case is the loss of your
friendship. In economic situations, the enforceable action has to be enforceable for such
cooperation to work. This might be some form of legal agreement, agreed punishment, fine etc.
Where such action cannot be enforced, the game becomes one of non-cooperation. I might not
care if I lose your friendship - presumably I would have considered that option as less valuable
than attracting the interest of the blonde!
What might be more interesting is, assuming our friendship was not totally destroyed, how the
'game' would be played the next time we went out together to try and acquire a date - would
you trust me to stick to our agreement this time? The outcome of such games, therefore, might
depend on the number of times that the games are played. We might go out together a number
of times and each time one of us betrays the other, but ultimately we might realise that this is
merely destroying our friendship and not actually getting us what we want. Cooperation may
well be the best option in the long term.
John Nash
It was into this arena that John Nash entered in the late 1940s and 1950s. Nash gained the
Nobel Prize for Economics for his contribution to game theory, which has had an impact on all
manner of economic and political situations. It is important to remember that Nash was not an
economist - he was a mathematician. In fact, Nash only took one course of economics during
his undergraduate studies - a course in International Trade - and that was only to fulfil the
degree requirements of the course he was on.
Nobel Prize winner for economics in 1994, John F. Nash Jr. Copyright: The Nobel Foundation.
Nash's ideas, therefore, are purely in the realm of the application of mathematical ideas to a
bargaining problem. The application to economics and other areas has been somewhat broader.
At the heart of Nash's ideas was the mix of both cooperative and non-cooperative games. In the
former, there are enforceable agreements between players whilst in the latter there are not. The
key thing in both cases is that the players in the game know that they cannot predict what the
other is going to do. Equally, they know what they want but are aware that all the other players
think as they do. It has been referred to as the 'I think he thinks that I think that he thinks that
I think...' scenario.
The solution/s that Nash derived were based around this thinking where each player had to try
to put him or herself in the position of others. The 'equilibrium' position would be where each
player makes a decision which represents the best outcome in response to what other players'
decisions are. The definition of a Nash equilibrium is a point where no player can improve their
position by selecting any other available strategy whilst others are also playing their best option
and not changing their strategies.
Let us attempt to illustrate this with an example:
Assume there are two firms competing with each other for profits in a market. The two firms
have three decisions to make with regard to their pricing strategies. They can choose to set
their price at either £10, £20 or £30. The matrix showing the profits made at different prices for
each is shown below.
Firm B
P = £10
P = £20
P = £30
P = £10 0; 0
60; -20
50; -30
P = £20 -20; 60
30; 30
100; 20
P = £30 -30; 50
20; 100
60; 60
Firm A
The matrix shows the level of profit or loss related to the three pricing decisions made by each
firm. If we look at the situation when each firm decides to set price at £30, they both make a
profit of 60. This, however, is not a Nash equilibrium, since Firm A could improve its position by
reducing its price by £10 to £20, whilst B's strategy remains the same. In this case, A would
now gain a profit of 100 rather than 60.
Let's compare this to the situation where both firms set price at £10. In this case, if Firm A
decided to raise price to £20, they would be worse off if B continued the strategy of charging
£10. In such an instance, A would make a loss of 20. The same position applies to firm B. There
is no incentive, therefore, for either firm to change their position; the payoff of zero profit at a
price of £10 represents a Nash Equilibrium.
Game Theory and Auctions
Nash's work in the field has been recognised as ground-breaking and it opened up the field for
application to a wide range of areas, including the biological sciences. Governments have
recognised the relevance of his work in helping them raise revenue from a variety of auction
type situations. This might be where the government are issuing licences for radio spectrum,
the rights to broadcast certain types of sporting event, licenses to mine for minerals or explore
for oil and most recently in the UK, the licences for operating 3G mobile phones.
Governments assemble experts from a given field to help them devise the structure of the
auctions, so that they can generate the most revenue. In some cases, the auctions have been
highly successful; in others, an abject failure. In one case where the auction failed, it was found
that the reason for the failure was that the players in the game were learning about the system
and 'cheating' by cooperating.
The auction for 3G mobile phones in the UK was planned with great care. It eventually raised a
staggering £22.47 billion. Original estimates suggested that it might bring in around £3 billion.
The auctions were characterised by having a high degree of perfect information between the
players - they could each see the others' bids. It appears that the mistake they all made was to
assume that the others knew something they did not and that the rising value of the bids
represented something of value - therefore, rival bidders felt the need to stay in the race.
Game theorists have since suggested that the bidding firms should have based their strategy on
the maximum amount they would be prepared to bid in relation to the business plan that they
had presumably set up
The auction for 3G licences in the UK raised over £22 billion. Maybe the players involved should
have read up on their game theory! Copyright: Drouu, from stock.xchng.
Game theory, therefore, has a direct relevance to our everyday lives in many ways. The growth
in the interest in this field is testament to the extent to which it has direct applicability to
economics and the real world. The work of people like John Nash has opened up new insights
into human behaviour and the way in which they interact, make decisions and engage in
economic bargaining and activity.