Focal point

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A focal point (also Schelling point or focal equilibrium ) represents a solution in game theory that the players choose when they cannot communicate with each other because this solution seems natural or outstanding to them. The concept was introduced in 1960 by Nobel Prize winner Thomas Schelling in his book The Strategy of Conflict .

Introductory example

Two businessmen agree on the phone that they want to meet in Paris at 12 noon . Suddenly the connection is broken without them being able to identify a more precise meeting point. Since the two wanted to meet at exactly 12 o'clock, they now have to find a point where they expect to meet the other because he also expects to meet them there. In Paris, for example, the Eiffel Tower can be chosen , not because it is a better meeting place than other places, but because it stands out among many possible solutions.

The successful solution depends on how well the business people already know their way around Paris. For someone more familiar with Paris, the Louvre , Arc de Triomphe, and Eiffel Tower can be three equally likely solutions.

Classification in game theory

The focal point is a coordination strategy in simultaneous, strategic games with several equilibrium situations (see Nash equilibrium ) and the requirement that all (or the majority) of the players simultaneously decide on the same equilibrium in order to win the game. In non-cooperative games, a Nash equilibrium is a stable solution, since no player has any motivation to deviate from the equilibrium once established.

Since no communication between the participants is allowed or possible in this game, it is important to guess which balance is assumed by all players as the expected result.

The expected equilibrium, the focal point, differs from all other equilibria in one outstanding property. Its selection is therefore more likely than the selection of another balance and it is therefore preferred by all players. As a result, it is not always the best balance, but often the outstanding balance that has the best opportunities.

Once a focal equilibrium has been established in the game, there is no better strategy for any player than the one that leads to this equilibrium outcome.

Importance of convention and context

The choice of focal balance is less often based on the rules of logic than on those of the prevailing conventions and the context of the players.

The context is described by the respective personal backgrounds, experiences and imaginations of all participants. The prevailing convention, i.e. the social and cultural norm, is also decisive for the choice of the focal result.

If players with similar contexts and conventions participate in the game, they will also prefer a common focal point. The more diverse the backgrounds of the players, the less likely they are to choose the same focal point.

The game theory includes cultural influences, at least possible equilibria, often made.

Application examples for focal equilibria

best result

If there are three Nash equilibria in the game with the payout amounts € 30, € 20 and € 10 for each player, all players will choose the strategies to receive the € 30 without voting. If a balance brings the best result for all players at the same time, this is automatically a focal point.

Sure result

Two players have the task of choosing between three Nash equilibria with the payout amounts € 10, € 20 and € 20. Only if they choose the same balance at the same time will they receive the respective payout. Without a match, they get nothing.

The payout of € 20 is the best result for both players. However, because there are two balances that pay off that amount, there is a risk in choosing. No player knows for sure which of the two equilibria the other will choose. Both get around this by opting for the balance with the € 10 payout.

In this example, it is not the best result, but rather the unique, safe result as a focal balance that leads to success.

Fair result

Two players in the same context are given the task of dividing an amount of € 10 independently of one another and without voting by determining their own share in whole euros. If both shares add up to € 10, the amount will be paid out in proportion. If the total is not € 10, no winnings will be paid out.

This game has eleven Nash equilibria: 0-10, 1-9, 2-8, 3-7, 4-6, 5-5, 6-4, 7-3, 8-2, 9-1, 10- 0

There is a high probability that both players will use a fair 50:50 strategy. This focal balance also leads to a Pareto-efficient result.

If the context of the players changes, other outcomes may be considered fair. For example, if a woman plays against a man, depending on the cultural area, a ratio of 40:60 or 20:80 can establish itself as a focal balance.

Unusual result

Each player in a group has the task of choosing a number from a sequence of numbers that he thinks will receive the most votes. If the player guesses correctly, he receives a payout of € 10, if he is wrong, there is no win.

Number sequence: 8, 12, 11, 16, 14, 15

Many players will choose the number 8 because it is the first number in the sequence and the only single digit number. The focal equilibrium lies with the number 8. If only mathematicians were to play, 11 would be even more unusual as the only prime number; the example also illustrates the importance of the common frame of interpretation (cultural context).

supporting documents

  1. ↑ a b cf. Avinash K. Dixit ; Skeath, Susan: Games of strategy; P. 109
  2. ↑ cf. John Maynard Keynes : The general theory; 7th edition; P. 156
  3. ↑ cf. Roger B. Myerson : Game theory - Analysis of conflict; P. 108
  4. ↑ a b cf. Avinash K. Dixit ; Skeath, Susan: Games of strategy; P. 110
  5. ↑ cf. Thomas Schelling : The Strategy of Conflict; P. 57
  6. ↑ cf. Avinash K. Dixit ; Barry J. Nalebuff: Thinking Strategically; P. 251
  7. ↑ cf. Roger B. Myerson : Game theory - Analysis of conflict; P. 114
  8. ↑ cf. Roger B. Myerson : Game theory - Analysis of conflict; P. 112

literature