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The Nash Equilibrium may be the most important idea in economics. Here’s why!

April 6, 2017

If there is a set of ‘game’ strategies with the property that no ‘player’ can benefit by changing their strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute what is known as the ‘Nash equilibrium’.

Assume each Player can adopt a ‘Friendly’ or a ‘Hostile’ approach to the game. For example, a Friendly strategy might be to put down the weapon you are carrying in your hand. The Hostile strategy is to keep hold of it.

Now, depending on their respective actions let’s say the game organiser awards monetary payoffs to each player. These are shown below and are known to each player.

Player B ‘Friendly’ Player B ‘Hostile’
   
Player A ‘Friendly’ 750 to A; 1000 to B 25 to A; 2000 to B
Player A ‘Hostile’ 1000 to A; 50 to B 30 to A; 51 to B

 

What is Player A’s best response to each of Player B’s actions?

If Player B acts ‘Friendly’, player A’s best payoff is if he acts ‘Hostile.’ This yields a payoff of 1000. If he had acted ‘Friendly’ he would have earned a payoff of only 750.

If Player B acts ‘Hostile’, player A’s best response is if he acts ‘Hostile. He earns 30 instead of a payoff of 25 if he acted ‘Friendly.’

In both cases his best response is to act ‘Hostile’. 

What is Player B’s best response to each of Player A’s actions?

If Player A acts ‘Friendly’, player B’s best payoff is if he acts ‘Hostile.’ This yields a payoff of 2000. If he had acted ‘Friendly’ he would have earned a payoff of only 1000.

If Player A acts ‘Hostile’, player B’s best response is if he acts ‘Hostile’. He earns 51 instead of a payoff of 50 if he acted ‘Friendly.’

In both cases his best response is to act ‘Hostile.’

Now, a Nash Equilibrium exists when Player B’s best response is the same as Player A’s best response. In this case, both Player A and Player B have the same best response to either action of his opponent. Both should act ‘Hostile’, in which case Player A wins 30 and Player B wins 51.

But if both had been able to communicate and reach a joint, enforceable decision, they would both presumably have acted ‘Friendly.’

Let’s now turn to the world of espionage in seeking out a Nash equilibrium. Let’s assume that there are two possible codes, and Agent A can select either of them and so can Agent B. The payoff to selecting non-matching codes is zero.

Agent B ‘Uses Code A’ Agent B ‘Uses Code B’
Agent A ‘Uses Code A’ 1000 to A; 500 to B 0 to A; 0 to B
Agent A ‘Uses Code B’ 0      to A; 0 to B 500 to A; 1000 to B

 

So where is the Nash equilibrium?

Top left: Neither Agent can increase their payoff by choosing a different action to the current one. In other words, there is no incentive for either Agent to switch given the strategy of the other Agent. So this is a Nash equilibrium.

Bottom right: As above

Top right:  By choosing to switch to Code B instead of code A, Agent A obtains a payoff of 500, given Agent B’s actions. Similarly for Agent B, who would gain by switching to code A, given Agent A’s strategy. So top right (Agent A uses code A and Agent B uses Code B) is NOT a Nash equilibrium, as both Agents have an incentive to switch given what the other Agent is doing.

Bottom left is the same as Top right. As above, there are incentives to switch. So it is NOT a Nash equilibrium.

In conclusion, this game has two Nash equilibria, top left (Agent A and Agent B both use Code A) and bottom right (Agent A and Agent B both use Code B).

Turning now to the classic Safe/Crash problem. In this problem if both drivers drive on the left of the road, they will be safe, while they will crash if one decides to adhere to one side of the road and the other to the opposite. This is shown in the box diagram below.

Driver B ‘ Drives on left’ Driver B ‘Drives on right’
Driver A ‘Drives on left’ Safe; Safe Crash; Crash
Driver A ‘Drives on right’ Crash; Crash Safe; Safe

 

So there are again two Nash equilibria here. Top left and Bottom right. In both these scenarios, there is no incentive for either Driver to switch to the other side of the road given the driving strategy of the other driver.

Now let’s consider the case of two companies who each have the option of using one of two emblems. We shall call the first the Blue Badger emblem and the other the Black Bull emblem.

Firm B uses Black Bull emblem Firm B uses Blue Badger emblem
Firm A uses Black Bullemblem 1000 to A, 500 to B 500 to A, 1000 to B
Firm A uses Blue Badger emblem 500 to A, 1000 to B 1000 to A, 500 to B

 

If we consider each section in turn, we arrive at the following result.

Top left: Firm B gains by switching from the Black Bull to the Blue Badger emblem.

Top right: Firm A gains by switching from the Black Bull to the Blue Badger emblem.

Bottom left: Firm A gains by switching from the Blue Badger to the Black Bull emblem.

Bottom right: Firm B gains by switching from the Blue Badger to the Black Bull emblem.

So this game has no Nash equilibrium.

So we have highlighted examples of games with one, two, and no Nash equilibria.

 

This leads us to the classic ‘Prisoner’s Dilemma’ problem. Are there any Nash equilibria here, and if so how many? In this scenario, two prisoners, linked to the same crime, are offered a discount on their prison terms for confessing if the other prisoner continues to deny it, in which case the other prisoner will receive a much stiffer sentence. However, they will both be better off if both deny the crime than if both confess to it. The problem each faces is that they can’t communicate and strike an enforceable deal. The box diagram below shows an example of the Prisoner’s Dilemma in action.

Prisoner 2 Confesses Prisoner 2 Denies
Prisoner 1 Confesses 2 years each Freedom for P1; 8 years for P2
Prisoner 1 Denies 8 years for P1; Freedom for P2 1  year each

 

The Nash Equilibrium is for both to confess, in which case they will both receive 2 years. But this is not the outcome they would have chosen if they could have agreed in advance to a mutually enforceable deal. In that case they would have chosen a scenario where both denied the crime and received 1 year each.

So, to summarise, a Nash equilibrium is a stable state that involves interacting participants in which none can gain by a change of strategy as long as the other participants remain unchanged. It is not necessarily the best outcome for the parties involved, but it is the outcome we would predict in a non-cooperative game of rational, self-interested actors.

 

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