From my doctoral course
This note comes from the doctoral course “Game Theory and Bargaining”. I rewrote the explanations in my own words and recomputed the numerical examples.
When I do not know who I am dealing with
In a game of incomplete information a player does not know what “type” the opponent is (ability, cost, preference). Harsanyi (1967) offered an elegant solution: nature first chooses each player’s “type” by a probability distribution, and each player sees only their own type. A game of incomplete information becomes a game of imperfect information and is solved with beliefs (probabilities).
Bayesian equilibrium
In a Bayesian Nash equilibrium each type of each player, given their beliefs about the opponent’s type, plays a best response. In dynamic games a perfect Bayesian equilibrium is also needed: beliefs are updated by Bayes’ rule after seeing moves.
Signaling
When the party with private information can take an action that reveals their type, signaling occurs. Michael Spence (1973) showed that education can be a signal: if education is cheaper for a capable worker, they show their ability by a degree and the market believes it, even if the degree adds nothing to productivity. Condition: the cost of the signal must be higher for the weak (otherwise everyone imitates).
Other examples: a warranty (a signal of quality), costly advertising, paying dividends.
Empty threats and credibility
A threat whose execution hurts the threatener is empty and, if the other side is rational, has no effect. To have effect it must become credible: either by prior commitment (losing the option of retreat) or by reputation (in a repeated game). That is why those who are seen as “irrational” in bargaining sometimes win more.
An example in negotiation
In Bargaining: power factors and the cost of waiting we saw that asymmetric information creates delay: the side with a high value must show by enduring delay that it really has that value, and the other side must distinguish between “rejecting the offer” and “a signal”.