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Claudia WinklemanYour browser does not support the element., a television presenter with a helmet of shiny hair, is not a typical economics teacher. Yet students should consider her game show. Those learning outside Britain may opt for any of the 20 or so versions of “The Traitors” screened elsewhere, including a popular American option that has featured celebrities such as Deontay Wilder, a boxing great, and John Bercow, a disgraced British parliamentarian. The game, which involves lying and betrayal, is a chance to study both the theory and reality of game theory, as well as to watch the panic on the face of someone who, having decided a fake Welsh accent would make them more trustworthy, comes across a native Welsh speaker.In game-theoretic terms, the show is a finite, sequential, incomplete information game, meaning it has an end, occurs in stages and players are kept in the dark about some things. A few are nominated by producers as “traitors”; the rest are “faithfuls”. The two groups live together. Each night the traitors pick a faithful to “murder”, who is then removed from the game. The next day the remaining players, both faithful and traitors, select a player to “banish”. Upon their exit, the banished player reveals whether they are a traitor or a faithful. The game continues for a set number of nights through to a final in which the last players have the option to repeatedly banish others until all remaining players declare themselves confident no more traitors remain. The survivors either split the prize—or hand it over to any undiscovered traitors.It is a variant of a game known as “Mafia”, which was invented in 1986 by Dimitry Davidoff, a psychology student at Moscow State University and secondary-school teacher on the side. Mr Davidoff invented the game for his pupils, hoping to demonstrate that an “informed minority” would triumph over an “uninformed majority”. They rarely do. He also aimed to show players would not only fail to identify the guilty but also confidently accuse the innocent, which does happen. The game spread by word of mouth until, in America, it was given gothic trappings of werewolves and villagers, rather than mafiosi, and became a staple at events in Silicon Valley. From a world-weary Soviet psychological experiment, it became a nerdy Californian parlour game and then, owing to a Dutch production company, mass tea-time entertainment.For a faithful, it is a fool’s errand to try to spot a lying traitor. Game theory identifies two types of communication: “cheap talk” and “signalling”. Saying you are “100% faithful”, as many contestants do, is cheap talk. It is costless and unverifiable: both a faithful and a traitor would make such a statement. Cheap talk is helpful only in a “co-ordination game”, where all players want the same outcome. Psychologists suggest the odds of telling whether someone is lying are little better than chance. As the game is finite—it is not repeated—there is no opportunity to learn any “tells”.There is also no chance for the more valuable kind of communication known as signalling. When signalling, a player takes a costly action in order to tell another player something. Some see a university education as an example of this: it costs cleverer and more conscientious types less to get a degree than stupider and lazier ones, allowing employers to distinguish between the two.With talk cheap, the only way to find a traitor is to study who is murdered and banished. One way of solving such a game is known as the “perfect Bayesian equilibrium”. Employing such an equilibrium, a player would calculate probabilities based on information revealed by behaviour: the chance of someone being a traitor depends on how likely a traitor would be to have taken their actions. In every “subgame” of the larger game, such a player would follow an optimal strategy based on these updating beliefs.Rational self-interest would suggest both faithfuls and traitors should turn on their allies in the final round: fewer to split the pot between. The very last game should consist of just three players who must choose to vote off one of their number (when the game gets down to two players no more voting can take place). A rational traitor should want to ensure this final trio consists of people who trust them. Yet that should lead any faithful to conclude that they have been kept in the game because they trust a traitor and have benefited from his or her protection. As a result, they should betray this person. The perfect Bayesian equilibrium, according to those who have studied Mafia, is voting randomly according to a pre-set public rule. Ensuring the rule is known to all players means that traitors who deviate and “just by chance” use their extra information to vote out only faithfuls are identified as doing so.Fortunately for television producers, contestants are not perfectly rational. Colin Camerer of the California Institute of Technology suggests that most players in actual games demonstrate “bounded rationality”, believing their strategy to be the most savvy, and responding to what they think less sophisticated players are doing. A “level zero” player might vote for who they think the traitor is; a “level one” might bluff; a “level two” might double-bluff, pretending to be unsophisticated. In cases where the skill of others is uncertain, cheap talk may sometimes be a useful strategy.Often the players simply protect those in their cliques and banish those whose behaviour is different. Abhijit Banerjee, a Nobel-prizewinning economist, developed a model of “rational herding”. Because behaviour provides information, and someone’s own information about the world is uncertain, it can be rational just to follow the crowd. Such decision-making may produce a feedback loop: as more people coalesce around an opinion, it becomes a better decision for everyone else to agree with it. For traitors this suggests an appealing strategy. Do not try to lead the discussion about who to banish, which might draw too much attention. Do enthusiastically agree with someone else’s mistake.
