Building Bayesian Games for Incomplete Information

The Anatomy of a Bayesian Game

In the last lesson, we saw how John Harsanyi's insight allows us to model incomplete information by introducing a "Nature" player. This framework, a Bayesian game, has a precise structure. Your background in Bayesian decision theory will be very helpful here, as the core of the model involves players maximizing their expected utility based on probabilistic beliefs.

To formalize this, a Bayesian game consists of a specific set of components. The following document provides a clear, formal definition of these elements.

To summarize and anchor these definitions, a Bayesian game requires us to specify:

• Players: The individuals making decisions (e.g., the hinge and a partner).
• States (Ω): The possible "realities" or states of the world that are uncertain. For instance, the true commitment level of a partner.
• Actions (A): The set of available moves for each player.
• Types (T): Each player has a "type," which corresponds to the private information they hold. A player's type is determined by the signal they receive from Nature about the true state of the world. An uninformed player has only one type.
• Beliefs (p): The probability distribution that an uninformed player holds over the possible types of the other players. This is their "prior" belief.
• Payoffs (u): A utility function for each player that depends on the actions of all players and the true state of the world.

Constructing a Bayesian Game for a Vee Scenario

Let's apply this structure to a classic vee relationship dilemma: assessing a new partner's intentions.

Scenario:
A Hinge (H) has been dating a new partner (P2) for a few months. The Hinge is uncertain about P2's long-term relationship goals. P2 knows their own goals but H does not. H believes that P2 is either a "Serious" type who desires deep integration and long-term commitment, or a "Casual" type who prefers a less entangled, more independent connection.

H must decide whether to make a "High Integration" move (e.g., suggesting they spend a major holiday together with H's other partner, P1) or a "Low Integration" move (e.g., suggesting a simple weekend date). P2, upon hearing the suggestion, can either Accept or Reject it.

Let's construct the Bayesian game step-by-step:

1. Players: Hinge (Player 1) and New Partner (P2, Player 2).

2. States: The true nature of P2's preference. Ω = {P2 is Serious, P2 is Casual}.

3. Types:

   • The Hinge has one type, as they have no private information in this scenario.
   • P2 has two types: $t_{P2}^S$ (Serious) and $t_{P2}^C$ (Casual). P2 knows their own type.

4. Beliefs: The Hinge believes P2 is the Serious type with probability $p$ and the Casual type with probability $1-p$. This is the common prior.

5. Actions:

   • Hinge: {High Integration, Low Integration}
   • P2: {Accept, Reject}

6. Payoffs: We must define plausible payoffs for each state. The numbers represent utility.

   • State 1: P2 is "Serious" ($t_{P2}^S$)
     The Serious type strongly prefers high integration.

     Hinge \ P2	Accept	Reject
     High Int.	(3, 3)	(0, 0)
     Low Int.	(1, 1)	(0, 0)

   • State 2: P2 is "Casual" ($t_{P2}^C$)
     The Casual type dislikes the pressure of high integration.

     Hinge \ P2	Accept	Reject
     High Int.	(0, -1)	(1, 1)
     Low Int.	(2, 2)	(0, 0)

This setup, with two separate game matrices, represents the situation from an omniscient perspective. But the Hinge doesn't have this view. They must make a choice based on their beliefs. To analyze this, we combine these two matrices into a single representation.

The Bayesian Normal Form

The standard method for analyzing this is to construct a single, larger game matrix where the informed player's strategies are contingent plans—one action for each possible type they could be.

The following video provides an excellent walkthrough of this exact process. It shows how to take two separate game matrices and combine them into one, calculating the expected payoffs along the way.

Following the method in the video, let's construct the "Bayesian normal form" for our vee scenario.

• Hinge's Strategies: {High Int., Low Int.}
• P2's Strategies: P2 has two types and two actions. Their strategy space is a set of plans. For example, the plan (Accept, Reject) means "Accept if I'm Serious type, Reject if I'm Casual type."
  • (Accept, Accept) -> A_S A_C
  • (Accept, Reject) -> A_S R_C
  • (Reject, Accept) -> R_S A_C
  • (Reject, Reject) -> R_S R_C

The resulting 2x4 matrix would look like this. The Hinge chooses a row, and P2 chooses a column.

Let's calculate the expected payoffs for the top-left cell: (High Int., (A_S, A_C)).

• Hinge's Expected Payoff (Eu_H):
  With probability $p$, P2 is Serious and H gets 3.
  With probability $(1-p)$, P2 is Casual and H gets 0.
  $E{u}_H=p(3)+(1-p)(0)=3p$
• P2's Expected Payoff (Eu_P2):
  This is also calculated from the Hinge's perspective.
  With probability $p$, P2 is Serious and gets 3.
  With probability $(1-p)$, P2 is Casual and gets -1.
  $E{u}_{P}2=p(3)+(1-p)(-1)=4p-1$

So, the payoff pair in the top-left cell is $(3p,4p-1)$. By repeating this calculation for all 8 cells, we have fully constructed the Bayesian game in its normal form, ready for equilibrium analysis.

The following resource provides several more worked examples of this construction process. You do not need to solve them all, but reviewing the setup of the "Gift Game" or "Two players in a dispute" will reinforce the procedure.

Conclusion

In this lesson, we established a formal procedure for taking a situation of incomplete information from a vee relationship and modeling it as a Bayesian game. This process transforms ambiguity and private knowledge into a concrete analytical structure.

Key Takeaways:

• Constructing a Bayesian game requires formally defining the players, states, actions, types, beliefs, and type-dependent payoffs.
• The key to analyzing such a game is to build its Bayesian normal form. This involves creating a single payoff matrix where the informed player's strategies are contingent plans (one action for each of their possible types).
• The payoffs for the uninformed player in this new matrix are their expected utilities, calculated by weighting the outcomes for each state by their beliefs (the prior probabilities).
• This construction allows us to use standard equilibrium analysis tools, like finding Nash equilibria, on a game of incomplete information.

We have now built the game. But what if the informed partner wants to reveal their type? How can a "Serious" partner credibly show their intentions to the Hinge, and how can the Hinge tell if this communication is genuine? In our next lesson, we will explore signaling games, which model precisely this kind of strategic communication.