Several players (who know they are a group of players) are told to pick individually a number between 0 and 100, which has to be close to 80% of the average number chosen by the group. The one who comes closest wins a monetary reward. What are the players supposed to do?
Game theorists say the following should happen. Some players might think the average number is 50 and pick 80% of 50, which is 40. Others might think that because of the above they have to pick 80% of 40, which is 32. Others might think that because of the above they have to pick 80% of 32, which is 25.6. This goes on and on … If the group is allowed to play continuously, the players will become fluent in meta-reasoning, i.e. fluent in thinking about thinking (theirs and other players’), and will all pick 0. In this particular instance, reaching 0 is reaching Nash equilibrium, a situation in which players no longer need to do what they have successfully done so far, i.e. no longer need to adjust their behavior with respect to other players' adjustments of behavior.
I have several questions:
1. What does a number between 0 and 100 mean?
Where exactly is the number: in (0, 100) or in [0, 100] or in (0, 100] or in [0,100)? This is an easy question if you have followed game theorists’ reasoning about reaching the Nash equilibrium. The number must be in [0, 100].
2. What does a number mean?
Are the players allowed to choose any number, φ or e for instance? It is natural to assume that numbers means whole numbers, but the reasoning of the game theorists shows that it is not the case (as 25.6 is not a whole number).
3. Is there only one Nash equilibrium if only whole numbers were allowed?
No, there is not. If only whole numbers were allowed, there are three Nash equilibria depending on the rounding algorithm. It is possible to round a number: a) up (to the higher whole number), b) down (to the lower whole number) or c) to the nearest whole number. If game organizers were true theorists they would give all the players an instruction how to round numbers.
If players were told to round down, then Nash equilibrium would be at 0. If they were told to round up the Nash equilibrium would be at 4. If they were told to round to the nearest whole number the Nash equilibrium would be at 2.
4. Is Nash equilibrium possible to achieve when rounding is not allowed?
The reasoning of the game theorists shows that rounding is not allowed. This is why a result such as 25.6 is admissible. This means that any positive number is an admissible answer. Then, Nash equilibrium is impossible to achieve, as 80% of any positive number, however small it is, is still a positive number (not 0). Zero is possible to achieve only if it is the first choice of all players, which is highly unlikely.
The weakness of Game Theory and Economics for that matter is that:
A. It is unreasonable to expect that ordinary people (i.e. the players of the economic game) will behave reasonably provided that those who are expected to be the most reasonable (i.e. those who set the rules of the economic game) do not behave reasonably (see above).
B. It is unreasonable to think there is always only one reasonable behavior. Any behavior is reasonable in the eye of the behaver, meaning that many different (even contradicting) behaviors are reasonable at the same time in the same part of space based on the individual circumstances.