Game Theory: The Nash Equilibrium Snappy Slides

What is Game Theory Game theory can be described as the study of the ways in which interacting choices of economic agents produce outcomes with respect to the preferences of those agents.

What is the Nash Equilibrium A key idea in the study of Game Theory is the Nash Equilibrium. The Nash Equilibrium, named after (for) John Nash, is a solution to a game involving two or more players who want the best outcome for themselves and must take account of the actions of others.

Nash Equilibrium Specifically, 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 the Nash equilibrium. Assume, for example, there is a game in which each Player can adopt a ‘Friendly’ or a ‘Hostile’ approach. So 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. An example of a payoff structure is shown in the next slide and is known to each player.

Nash Equilibrium – Step by step guide. Step 1: Here is the payoff matrix: 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

Nash Equilibrium – Step 2 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’.

Nash Equilibrium – Step 3 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.’

Nash Equilibrium – Step 4 Now, a Nash Equilibrium exists when Player B’s best response is the same as Player A’s best response. 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.’

Nash and the Spy Game Let’s turn now 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. An example of a payoff structure is shown in the next slide and is known to each Agent.

Find the Nash equilibrium Player B ‘Uses Code A’ Player B ‘Uses Code B’ Player A ‘Uses Code A’ 1000 to A; 500 to B 0 to A; 0 to B Player A ‘Uses Code 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. So there is no incentive for either Agent to switch given the strategy of the other Agent. So this is a Nash equilibrium. Bottom right: Same as Top left. This is a Nash equilibrium. Top right: By choosing to use 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 this box (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: Same as Top right. As above, there are incentives to switch. So it is NOT a Nash equilibrium. Conclusion: This game has two Nash equilibria, top left (both Agents use code A) and bottom right (both Agents use code B).

Live or Die? The Car Crash Problem. 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, whilst 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 next slide.

Find the Nash Equilibrium Player B ‘ Drives on left’ Player B ‘Drives on right’ Player A ‘Drives on left’ Safe; Safe Crash; Crash Player A ‘Drives on right’

Solution to the Car Crash Problem At Top left and at Bottom right, there is no incentive for either Driver to switch to the other side of the road given the driving strategy of the other driver. So both Top left and Bottom right are Nash equilibria. In both other scenarios (Top right and Bottom left), there is a very strong incentive to switch to the other side given the driving strategy of the other Driver. So neither Top right nor Bottom left is a Nash equilibrium. In summary, there are two Nash equilibria in the Car crash problem.

The Clash of the Company Emblems 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.

Find the Nash equilibrium Firm B uses Black Bull emblem Firm B uses Blue Badger emblem Firm A uses Black Bull emblem 1000 to A, 500 to B 500 to A, 1000 to B Firm A uses Blue Badger emblem

Is there a Nash equilibrium? Top left: Firm B gains by switching from Black Bull to Blue Badger Top right: Firm A gains by switching from Black Bull to Blue Badger Bottom left: Firm A gains by switching from Blue Badger to Black Bull Bottom right: Firm B gains by switching Blue Badger to Black Bull So this game has no Nash equilibrium. There is always an incentive to switch.

So how many Nash equilibria can there be? There may be one (e.g. the Friendly/Hostile game) There may be more than one (e.g. Spy problem, ‘Live or Die’ problem) There may be none (e.g. company emblems problem)

The Prisoner’s Dilemma problem This leads us to the classic ‘Prisoner’s Dilemma’ problem. In this problem, 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 next slide shows an example of the Prisoner’s Dilemma in action.

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

Prisoner’s Dilemma – The Outcome 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.

Nash Equilibrium - Summary So 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.