Relations between three types of Equilibrium
In class, we discussed about the way that Nash Equilibrium is shown in some two-player games, for example, football games and prisoner dilemma. Besides Nash Equilibrium, there also exist Bayesian Nash equilibrium and Perfect Bayesian Nash equilibrium. In this blog post, we will explore the relationship among the three kinds of equilibrium.
In the cases we analyze in class, I found out that Nash Equilibrium is actually a proper state from which neither of the two players in the game wants to be deviated. In this situation, the two players consider more as a group than as individuals. However, the situation is different in Bayesian Nash equilibrium. We can see more subjectivity in Bayesian Nash equilibrium, which means the two players’ believes are more likely to be considered subjectively. That’s the reason that we can find Bayesian Nash equilibrium in first-price auction (as shown in the article below). In the Bayesian Nash equilibrium, the two players are usually of different types, which makes them hard to predict other’s choice. The two bidder don’t have to consider other bidder’s strategy (They do not know other’s strategy.), instead they choose the strategy which gains themselves the best value. Therefore, there does not exist the Dominating Strategy for first-price auction.
There is another example for Bayesian Nash Equilibrium. In an office which has two types of workers: A and B. Let us suppose A and B has the same original productivity. Now, A and B hold the “belief” that A’s productivity is less than that of B, and they work as what the “belief” says. Then, since A works with less productivity, the employer paid less to A than did to B:
A: Less Productivity; Less Pay
B: More Productivity; More Pay
In another perspective, working with less productivity is the best strategy for A, for A got less pay from the employer. This example could be considered as Perfect Bayesian Nash equilibrium, in which players’ “belief” and their strategies work together to form a dynamic equilibrium.
From the example above, we can tell that Perfect Bayesian Nash equilibrium is the most limited version of Nash equilibrium. The relations of three Equilibrium is shown in the picture.
Source:http://www.econport.org/content/handbook/auctions/auctionsexperiments/auctionsbneandfirstprice.html