What is the point of considering only pure strategies in a game? How could you restrict people from thinking about mixed strategy?

In an experimental setting, how could you effectively incentivize the subjects to not to adopt mixed strategy?

I would like to re-emphasize that the question in concern is "how to prevent people from using mixed strategy" such that only pure strategies are adopted. The mixed strategies must be theoretically adoptable, and we have a mechanism to force people to think about pure strategies only. In a non-repeated game, if a person plays strategy "H", in general you don't know if it is pure strategy "H" or a mixed strategy with positive probability on "H". The current answers are very useful and well-prepared; however, what I am always looking for is a proven (either theoretically or experimentally) method that constrains the choice set from a mixture space to a doublet.

Technically, in a game theory experiment, the set of alternatives is a mixture set. I want to restrict the set of alternative to two objects, $$\{H, T\}$$, only.

Of course, you could just post a title of a paper which includes an incentive mechanism or an experimental design. This can be a perfect answer despite of its length. Opinions are welcome but opinions are not answers.

Let's consider a one-period game where the first player choose $$H$$ or $$T$$. The game satisfies the following two conditions:

1. The equilibrium is unique at $$\frac{2}{3} H+\frac 1 3 T$$.

2. If the player is restricted from choosing mixed strategy, $$T$$ becomes the optimal choice for player 1.

I've chatted with a few people and they all believe that it makes sense to consider those two cases separately; by "separately" they mean that, a game problem restricting the player from choosing mixed strategy also make sense, we can compare $$H$$ and $$T$$ in isolation of those mixed strategies. What is the philosophy behind this?

I, on the other hand, believe that the latter case doesn't make sense in real life: one cannot just abandon the mixed strategy completely. In an experimental setting, how could you prevent the players from adopting a mixed strategy?

One awkward way to impose this restriction, I think, is to instruct the player that it is a super game repeated for 10 times (or even life-time); you can only choose the same $$H$$ or $$T$$ for the lifetime and never change your choice. But this way, people can still think about mixed strategy.

Clarification: I am not advocating that people must use mixed strategy only. I am just saying that, I cannot find a good experiment that restricts people from including mixed strategies in their menu. Similarly, I cannot find a good experiment that restricts people from including pure strategies in their choice sets. So I think, in analysis, we must think about all strategies together, and it is pointless to consider pure or mixed strategies only.

PS: Assume the player is rational, as this is economics SE.

3 Answers

If in equilibrium, a player "chooses a mixed strategy" that plays $$H$$ and $$T$$ with positive probability, $$H$$, and $$T$$ must be both optimal choices. It is a standard result that for a (subjective or objective) expected utility maximizer, randomizing can only be optimal if it is over pure optimal choices. This is a direct consequence of expected utilities being linear in probabilities. So the problem you mentioned can never occur.

So expected utility maximizers do not ever have a strict incentive to randomize, which raises the question of how one can interpret Nash equilibria in mixed strategies. One popular interpretation is that a player's mixed strategy is really representing a shared probabilistic belief other players have over the choices of that player. One can then define Nash equilibrium as a consistency condition on beliefs, though it is not clear why this condition should obtain in practice. Oner idea, Harsanyi's purification idea is that with a certain probability the player has private information that makes $$H$$ or $$T$$ uniquely optimal, but for the other players it will look like choosing $$H$$ and $$T$$ randomly. Especially in zero-sum games, there is also the idea that players deliberately use randomization devices such as coins so that their plans cannot be found out. Lastly, there is the "mass action" interpretation that can already be found in Nash's thesis. According to this interpretation, you are randomly matched with players from a huge population that play a fixed pure strategy, and the mixing probabilities represent the population fractions playing each pure strategy.

• +1 In the first paragraph, are you saying that people cannot have $H\succ T$ or $T\succ H$ if the mixed strategy is the sole Nash Equilibrium? Commented Oct 14, 2020 at 9:26
• Yes. Both $H$ and $T$ must be optimal against whatever the other player is "playing" in equilibrium. That is also how we calculate equilibria in mixed strategies, we find the probabilities that make the opponent indifferent. Commented Oct 14, 2020 at 9:29
• Is it true that researchers usually find pure strategies first, and then, if the pure strategies NE does not exist, people will turn to a mixed one? I found this sentence on Wikipedia without sources: "Mixed strategies are still widely used for their capacity to provide Nash equilibria in games where no equilibrium in pure strategies exists, but the model does not specify why and how players randomize their decisions." Commented Oct 14, 2020 at 9:54
• It depends on what you are looking for. If you want to find all equilibria, you need to look for mixed equilibria even if you found pure ones. Commented Oct 14, 2020 at 10:10
• That is certainly what many people do in applied work. And while it is hard to formulate general principles that explain this, I would not expect people deciding non-cooperatively on which side of the street to drive to eventually settle on the mixed-strategy equilibrium. Commented Oct 14, 2020 at 10:26

The real-life question is "how do you persuade people to use mixed strategies"?

To stick with your example, Consider a person that has to make a binary choice $$(H, T)$$, and, after contemplation, they conclude that the optimal strategy is the mixed strategy $$(2/3, 1/3)$$. I have never know of anyone putting two red and one blue ball in a vase and then picking randomly in order to make the decision. Rather, they choose the $$2/3$$ strategy.

Most people don't like chance, and certainly don't like to explicitly "leave decisions to chance". It is no accident that "flipping a coin to decide" is always mentioned with a shrug, in recognition of our inability to move away from a 50-50 split. We don't like the 50-50 split, it creates a dilemma, which always has negative connotations. The implication is that we would like to be always away from 50-50 so that we didn't have to flip a coin. It follows that when we are away from 50-50 we take the highest weighted option with relief, because we then feel that we are not leaving the decision to chance. So, once more

How do you persuade people to use mixed strategies as they should?

• As evident in many psychology and economics experiments, people do naturally use mixed strategy, even if it a suboptimal mixed strategy. Commented Oct 13, 2020 at 1:23
• I think you made a good point, but did not addressed to the question. In the question, I never dismiss pure strategies. My point is, you cannot dismiss either mixed or pure strategy. They must come together. I never said people must always use mixed strategy; I am just saying that people can use mixed strategy and you cannot really design an experiment to prevent the possibility of using mixed strategy, just like you cannot design an experiment to prevent people from using pure strategies. Commented Oct 13, 2020 at 1:28
• 1) In real life, most time they don't. Just shop around and ask them how many times did they put the red and the blue balls in a vase and drew blindly, to actually make a real life decision 2) The purpose of my post was to explain why a game constrained to use only mixed strategies is also useful: because it reflects accurately many-many real world situations. So I did address the question, convincingly or not. Commented Oct 13, 2020 at 2:13
• People did not intentionally use a mixed strategy doesn't mean that they don't use mixed strategies subconsciously. In fact, in game theory and decision theory there are overwhelming evidence that people intentionally or subconsciously randomize, which should not be confused with risk-aversion. Commented Oct 13, 2020 at 2:20
• How do you play Rock/Paper/Scissors? Do you always choose rock, always choose paper, or always choose scissors? Or, perhaps, does picking one from a vase result in a better outcome? Commented Oct 14, 2020 at 13:43

In an experimental setting, how could you prevent the players from adopting a mixed strategy?

I don't think you can. Restricting access to mixed strategies is essentially banning the use of any private randomization devices. But since there are various ways to perform mental coin-flips, not all of which are readily observable, it would be prohibitively difficult to control for the use of private randomization devices, and hence mixed strategies, even in a lab setting.

Nevertheless, you can infer from a subject's choice data whether or not they adopted only a pure strategy in a game. A commonly used trick in economic experiments is repeated play with strangers and with no feedback. That is, a subject plays the same game against a series of new players (with whom they only interact once) and they don't see the results of these interactions until the end of the session. Re-matching with strangers and providing no feedback controls (imperfectly) learning during the play, which allows the experimenter to reasonably assume independence across the repetitions. If a player choses the same action in each repetition, then it's probable that they've used a pure strategy, among a set of possibly mixed strategies. This is still not the same as restricting choice to only pure strategies though.

• Can't they choose a pure strategy at random and then use the selected pure strategy against each partner. For all purposes, that is pretty much playing a pure strategy. Commented Mar 31, 2021 at 21:07
• @MichaelGreinecker: Yes they could. We just won't be able to assert with the same level of confidence that the observed choices were due to the player playing different pure strategies in different rounds. Note that even in the case I described, the conclusion is probable: the same action across all rounds could just be a sequence of (highly unlikely) outcomes of a mixed strategy. Commented Mar 31, 2021 at 23:35