Bayesian form games, situations in real life for which having more information is not necessarily better

Question

I was given these two bimatrices, for two different versions of a Bayesian-form game.

In the first version of the game, a player named The Nature chooses between $A$ or $B$ with probability $1/2$ each. The players don't know what The Nature chose. After that, players 1 and 2 play simultaneously, with utilities given by the bimatrix on the left if The Nature chooses $A$, or the right one if The Nature chooses $B$. In this game I found a Nash equilibrium when player 1 chooses $Z$ and player 2 chooses $V$, with expected utilities $5$ and $1$, respectively.

In the second version of the game, player 1 knows what The Nature played, but player 2 doesn’t. Then they play the respective game simultaneously. Here I found a Nash equilibrium when player 1 chooses $(X_A, Y_B)$, and player 2 chooses W, i.e. if The Nature plays $A$ player 1 chooses $X$, and if The Nature plays $B$, player 2 chooses $Y$. However here the expected utilities of the players are $4$ and $1$ respectively, which seems counterintuitive since player 1 had more information, but gets a worst expected utility.

I want to know why this might be the case, and examples of games in real life (not necessarily related to economics) where this kind of things may happen.

Answer 1

having more information is not necessarily better

Part of the reason for this outcome is that player 2 knows player 1 has more information (than player 2 himself) and therefore chooses a different strategy than in the case where neither player observes the state. If somehow player 2 is ignorant of player 1's information advantage and mistakenly believes that player 1 is as uninformed as himself, then player 1 would be able to get 6 in each state. But of course this cannot be an equilibrium. Thus, the asymmetry of information is as crucial as the amount of information in determining equilibrium outcomes.

There have been theoretical and empirical work that confirm the quoted result above. For example,

• Green (1981). Value of Information with Sequential Futures Markets

• Hirshleifer (1971). The Private and Social Value of Information and the Reward to Inventive Activity

• Sharma (2017). Is more information always better? A case in credit markets

• Huber et al. (2008). Is more information always better?: Experimental financial markets with cumulative information