When Should We Cooperate, and When Should We Betray?

Lessons from Game Theory, and the Traitors.

A version of this article can be found in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press).

The formal foundation for modern game theory was laid by John von Neumann and Oskar Morgenstern in the 1940s, with their ground-breaking book, Theory of Games and Economic Behavior. This work sets the stage for game theory as a crucial analytical tool in economics. In the 1950s, John Nash extended its scope, introducing concepts like the Nash Equilibrium. These milestones marked game theory’s evolution from a mathematical curiosity to a pivotal tool in various disciplines.

Essentially, game theory focuses on examining how rational individuals (or entities, which could for example be groups, organisations, or countries) make decisions to maximise their own outcomes in situations where the decisions of one so-called ‘player’ affect the outcomes of other ‘players’, and vice versa. It provides a framework to analyse and predict the choices of rational actors in these strategic situations.

APPLICATIONS IN VARIOUS FIELDS

The versatility of game theory is evident in its wide-ranging applications across multiple fields. It can be applied to a wide range of scenarios, from everyday decision-making and business strategy to complex negotiations and interactions in international relations. In economics, it models market dynamics, helping understand competitive strategies and the dynamics of auctions. Biologists use game theory to analyse animal behaviour, such as mating rituals and foraging strategies, viewing them as strategic games for survival. In political science, game theory is applied to electoral strategies, voting systems, and international diplomacy, providing insights into the strategic behaviour of voters, politicians, and nations.

UNDERSTANDING THE NASH EQUILIBRIUM

An essential concept in game theory is the Nash Equilibrium. This represents a state in which, given the strategies of the other players, each player’s chosen strategy maximises their payoff and they have no incentive to deviate from it. In other words, a Nash Equilibrium occurs when all players choose the best response to the other players’ strategies.

In a game reaching a Nash Equilibrium, no player can unilaterally improve their situation by changing their strategy, assuming other players’ strategies remain fixed.

Take the case of two spies, Anna and Barbara: if they both use the same code, they successfully communicate and gain a reward; if they use different codes, the communication fails, resulting in no payoff. Here, the Nash Equilibria are the scenarios where both spies choose the same code—either code 1 or code 2. In these situations, no player can improve her payoff by unilaterally changing her strategy given the strategy of the other.

Similarly, two drivers must decide which side of the road to drive on. Assuming that their concern is to avoid a collision, the Nash Equilibria occur when both drivers choose to drive on the same side, either left or right. It is beneficial for each driver to imitate the choice of the other; if one driver deviates, it would result in a collision. In this case, therefore, there are two Nash Equilibria—both drive on the left or both drive on the right.

In other situations, no stable state or Nash Equilibrium can be achieved. Take as an example two companies choosing a company logo. If Company A chooses a bear logo, Company B may have an incentive to switch its own logo to a bull. However, if it does so, Company A may now have an incentive to change its logo again, and so on. This situation has no Nash Equilibrium, where a Nash Equilibrium means no company would benefit by changing its strategy given the strategy of the other.

These examples show how the concept of the Nash Equilibrium is applied in different contexts to predict and analyse strategic choices. However, they also underscore the complexities of game theory and strategic interactions. Some games might not have a Nash Equilibrium, and in others, there may be multiple equilibria.

The concept of the Nash Equilibrium becomes more interesting and complex in larger games with more players and more possible strategies. For instance, in the marketplace, firms’ pricing strategies could reach a Nash Equilibrium where no single firm could increase its profit by unilaterally changing its price, given the prices of its competitors. An example is a simple two-firm scenario where both firms know that if either raised their price, they would lose most or all of their customers to the other firm.

However, it’s important to note that Nash Equilibria do not always lead to the best collective outcome. The classic example is the ‘Prisoner’s Dilemma’, where the Nash Equilibrium strategy for both players leads to a worse collective outcome than if they could collaborate. This highlights that while the Nash Equilibrium is a powerful concept in understanding strategic behaviour, it is not necessarily synonymous with individual or group optimality.

UNDERSTANDING THE PRISONER’S DILEMMA

The Prisoner’s Dilemma is a classic problem in game theory, illustrating why it can be hard for rational individuals to cooperate, even when it is in their best interest. In this scenario, two prisoners are individually given the option to confess or deny a crime they have committed together. Depending on the combination of their decisions, they can either reduce their sentences, remain the same, or one can go free while the other gets a heavier sentence. They cannot communicate or collude.

If one confesses and the other doesn’t, the prisoner who confesses is released. If both confess, each is better off than denying while the other confesses. The Nash Equilibrium in this game is for both prisoners to confess, which is not the optimal outcome for either. This problem illustrates a situation where individuals’ rational decisions can lead to a collectively undesirable outcome.

Let’s demonstrate this with an example:

Imagine two prisoners who are part of the same crime. They can’t talk to each other. Here is their dilemma:

If both confess, they each get two years in jail.

If one confesses and the other denies, the one who confesses goes free and the other gets eight years.

If both deny, they each get only one year in jail.

The smartest move for two self-interested prisoners is in each case to confess, because they can’t be sure what the other will do. This is the Nash Equilibrium. But if they could make a deal, they would both deny the crime and get just one year each.

This situation also shows a ‘dominant strategy’, where the best choice (confessing) doesn’t depend on what the other person does. It’s the best move no matter what.

But not all situations have a dominant strategy. Take driving on the right or left side of the road. In the US, driving on the right is the norm, so it’s best for everyone to do that. In the UK, it’s driving on the left. These are examples of Nash Equilibria, where everyone’s doing what is best considering what others are doing.

So, a Nash Equilibrium is a stable situation where nobody gains by changing their strategy if others don’t change theirs. It’s not always the best for everyone involved, but it’s often what happens, especially among rational, self-interested people. Sometimes the best strategy in theory is not the best in practice.

GOLDEN BALLS DILEMMA

An example of the Prisoner’s Dilemma in action is a one-time TV show called ‘Golden Balls’ where two players each choose a ball—either ‘Split’ or ‘Steal’, without knowing what the other chooses. They can talk before choosing. Here’s what happens next:

If both choose ‘Split’, they share the prize money equally.

If both choose ‘Steal’, neither gets any money.

If one chooses ‘Steal’ and the other ‘Split’, the ‘Steal’ player gets all the money, and the ‘Split’ player gets nothing.

In this game, the best bet for self-interested players (as in the Prisoner’s Dilemma) is to both choose ‘Steal’, because choosing ‘Steal’ is always better or no worse than choosing ‘Split’. ‘Steal’ in this game is like ‘Confess’ in the Prisoner’s Dilemma.

The difference is that in the Prisoner’s Dilemma, the players can’t talk to each other. In Golden Balls, they can. They could both win half the prize if they agree to ‘Split’, but they risk losing everything if they both choose ‘Steal’. The show often has players agreeing to ‘Split’, but then one or both betray the agreement and pick ‘Steal’.

This demonstrates that even when players can talk and agree in games like these, they can still end up not cooperating if there’s no way to enforce their agreement. Indeed, the more credible is the promise to split, the more tempting it may be for the opponent to steal. This tells us that not even communication and agreement can resolve the Prisoner’s Dilemma when there is no way to enforce the agreement. This ‘problem of credible commitment’ is a common feature of many strategic interactions in real life.

TRAITOR’S DILEMMA

In some versions of the reality TV show “The Traitors” there is a variant of the Golden Balls Dilemma but with more players, called the Traitor’s Dilemma.

To analyse this, we can consider the different scenarios and the rewards associated with each. The outcomes depend in this example on the decisions made by three players:

If all decide to share, they each get one-third of the pot.

If all decide to steal, they all get nothing.

If two decide to steal and one decides to share, the two who chose to steal split the pot (each getting half).

If one decides to steal and the others decide to share, the one who chose to steal takes the whole pot.

First, notice that if a player expects the other two to Share, their best response is to Steal, since this would give them the entire pot instead of just one third. If a player expects one to Steal and one to Share, their best response is also to Steal, ensuring they get at least half the pot (if two steal) instead of nothing. If a player expects both others to Steal, their best strategy is indifferent between Stealing and Sharing since both result in no prize; however, in typical game-theoretical analysis, such players might lean towards Stealing out of self-interest, as it does not worsen their situation but has a potential benefit.

Thus, we can infer that in each situation, choosing to Steal can never result in a worse outcome for a player than choosing to Share, assuming the other players’ actions are fixed.

However, this equilibrium is precarious in real-world contexts, especially if the players can communicate or have formed trust throughout the game, as mutual cooperation (all choosing to Share) leads to a better outcome for the group compared to the individual rationality of Stealing.

So, in the context of game theory, the optimal strategy in a single shot of this game, without considering trust or external factors, would be to Steal, as it maximises the player’s minimum possible gain, given the assumptions typical in game theory of rationality and self-interest. However, “optimal” can vary based on the context of previous rounds, relationships, or possible future repercussions outside the standard game-theoretical framework.

ITERATED GAMES AND REPUTATION

One solution to the problem of credible commitment is through the concept of iterated games and the development of a reputation. An iterated game is a repeated version of a basic game. In these games, players can observe the actions of their opponents over multiple rounds, allowing them to adjust their strategies based on their opponents’ past behaviour.

In the context of our ‘Golden Balls’ or ‘Traitors’ examples, if the game were to be played repeatedly a player who breaks their promises would quickly gain a reputation for being untrustworthy. Knowing this, they would be more likely to keep their promises to maintain their reputation and the trust of their opponents.

THE DOLLAR AUCTION

The ‘Dollar Auction’ paradox relates to a scenario where Mr. Moneymaker auctions off dollar bills. The rule is that the highest bidder wins the dollar, but the second-highest bidder also must pay their bid and gets nothing. Here’s an example:

Someone bids 1 cent, hoping to make 99 cents profit.

Then another bids 2 cents, and so on, up to 99 cents.

At 99 cents, the person who bid 98 cents doesn’t want to lose that money, so they bid $1.

This keeps going, with each bidder trying to avoid losing their bid amount. It becomes a cycle where the only winner is Mr. Moneymaker, the auctioneer.

This example, along with the previous ones, highlights the successes and failures of communication and coordination. Finally, let’s touch on ‘focal points’ or ‘Schelling points’. These are strategies people naturally pick to coordinate without communication. An example is when people were asked to meet a stranger in New York City without any specific instructions or prior communication. Many chose 12 noon at Grand Central Station as the meeting point. This is a ‘Schelling point’ because it’s a natural and obvious choice for coordination in the absence of communication.

CONCLUSION: GAME THEORY— A POWERFUL TOOL

The Prisoner’s Dilemma, Nash Equilibrium, and the broader field of game theory provide powerful tools for analysing situations of strategic interaction. While these tools can highlight potential outcomes and strategies, they also expose the inherent challenges involved in these situations, such as the problem of credible commitment and the importance of reputation in iterated games. By understanding these concepts, we can better understand the complex dynamics of many real-world scenarios, from business negotiations to international politics.