Welcome back. In our last lesson, we analyzed how the perceived duration of a relationship profoundly affects strategic cooperation. We saw that for cooperation to be sustainable, there must be an "indefinite future." This entire analysis, however, rested on a crucial assumption: that all partners share the same perception of the relationship's future. What happens when this, or other critical information, is not common knowledge?
This lesson marks the beginning of our fourth module, "Information Asymmetry and Signaling." We will now move from situations where players have complete information to the more realistic and complex world of incomplete information. Your learning outcome for today is to model asymmetric information scenarios in a vee where one or more partners hold private information. We will explore the formal game-theoretic tools used to represent situations where one person knows something that others do not—a defining feature of human relationships.
When You Don't Know Who You're Playing With
In all the games we've considered so far, we've assumed that players know everything about the structure of the game: who the other players are, their available strategies, and, most importantly, their payoffs. When players don't know the moves others have made in a simultaneous-move game, we call this imperfect information.
But a more fundamental uncertainty often exists. What if you don't fully know the other players' preferences or motivations? This is a game of incomplete information. One partner might have private knowledge of their own commitment level, their outside options, or their true feelings about a situation. This is not just uncertainty about an action; it's uncertainty about the very payoffs that drive the game.
The following presentation slides provide a clear distinction between these concepts and introduce the foundational method for analyzing games of incomplete information.
4.4 — Incomplete & Asymmetric Info. - Game Theory
This resource introduces the key concepts of imperfect vs. incomplete information and outlines the Harsanyi transformation, which is the standard approach for modeling these games.
Please read the first few sections of the slides, focusing on these parts: Under "Information in Games", review the definitions of perfect, imperfect, and incomplete information. Examine the concept of an information set, which we use to model imperfect information in extensive form. Read the section on "Incomplete Information" and how we introduce Nature to model external uncertainty. Finally, read the introduction to Asymmetric Information and the Harsanyi transformation.
The breakthrough insight, developed by Nobel laureate John Harsanyi, is that any game of incomplete information can be converted into an equivalent game of complete but imperfect information. We do this by introducing a fictional player called "Nature."
Here's how it works:
- Types: Instead of saying a player's payoffs are unknown, we say there are different possible "types" of that player, each with a different set of payoffs. For example, a non-hinge partner might be a "High Commitment" type or a "Low Commitment" type.
- Nature's Move: At the beginning of the game, "Nature" makes a random move that determines which type each player is.
- Beliefs: The other players don't observe Nature's move, so they don't know the player's actual type. However, they have beliefs, represented by a probability distribution, over the possible types. For instance, a hinge might believe there is a 70% chance their new partner is the "High Commitment" type and a 30% chance they are the "Low Commitment" type.
The game is now one of imperfect information because players know the full structure (all possible types and payoffs) but are uncertain about Nature's initial move. This framework allows us to analyze the situation using familiar tools.
Bayesian Games and Bayesian Nash Equilibrium
This transformed game is called a Bayesian game. To solve it, we need an equilibrium concept that accounts for players' beliefs.
This short video explains the key additions for a Bayesian game and defines the corresponding equilibrium concept.
Game Theory 101 (#64): Bayesian Nash Equilibrium
This video by William Spaniel defines the components of a game with incomplete information and introduces Bayesian Nash Equilibrium.
First, watch the segment explaining the new elements: types and beliefs. Next, watch the definition of Bayesian Nash Equilibrium (BNE).
As the video explains, a Bayesian Nash Equilibrium (BNE) is a set of strategies, one for each type of player, such that no type has an incentive to unilaterally change their strategy, given their beliefs about the other players' types and the strategies played by those other types.
In a BNE, an uninformed player's strategy maximizes their expected utility. Given your background in statistics, you'll recognize this as a standard decision-theoretic calculation: the player weighs the payoff of each possible outcome by the probability (their belief) that this outcome will occur, and chooses the action that yields the highest weighted average.
The slides you read earlier provide a step-by-step example of finding the BNE in a game where one player ("Rowena") is uncertain about the other player's ("Colin's") type.
4.4 — Incomplete & Asymmetric Info. - Game Theory
This part of the presentation walks through a concrete example.
Please review the section titled "Bayesian Game Example". Follow the logic for how Rowena calculates her expected payoff for each action and how the equilibrium depends on her beliefs (the probability p).
Modeling Asymmetric Communication in a Vee
Let's now apply this framework to a highly relevant scenario in a vee relationship: private conversations. Imagine a vee with a hinge (H), an established partner (P1), and a newer partner (P2). When H and P1 have a private conversation, P2 knows they talked but doesn't know the content. This creates asymmetric information. Did they discuss something that benefits the whole vee, or did they form a "coalition" to pursue their own interests, possibly at P2's expense?
This exact situation is modeled in a fascinating experimental paper. The setup involves a three-player game where two "partners" can communicate privately, excluding a "lone person." This structure serves as an excellent formal model for the dynamics of a vee with private communication channels.
[PDF] Asymmetric communication in a three-person dilemma - EconStor
This paper provides an experimental model of a three-player game with asymmetric information, directly analogous to a vee relationship.
Please read the following sections to understand how the authors model this scenario: The Abstract on the first page provides a concise summary of the setup and key findings. The Introduction (pages 1-2) frames the problem, highlighting how exclusionary communication can be harmful. The section on Basic Setup and Theory Prediction (pages 7-8) describes the underlying investment and allocation game. Pay close attention to the description of the "full investment equilibrium" versus the "collusion equilibrium." Finally, read Private (Exclusionary) Communication Treatments (pages 8-9) to see how private information is introduced through a private chat channel.
Let's dissect this model in the context of our vee:
- Players: The two "partners" (A and B) map to the Hinge and P1. The "lone person" (C) maps to P2, the newer partner who is excluded from the private conversation.
- Asymmetric Information: The private communication channel creates two "types" of players: the informed pair (H and P1) and the uninformed individual (P2). P2 knows a conversation occurred (they see the "hashtags" in the experiment) but is uncertain about its content.
- Beliefs & Strategic Response: P2 must form beliefs about what H and P1 likely discussed. If P2 believes there's a high probability of a "collusion equilibrium"—where H and P1 will act in their mutual interest and exclude P2 from the benefits—P2's best response is to protect themselves by withholding their own contribution (investing zero). As the paper's results show, this is precisely what happens. The mere existence of an exclusionary communication channel is enough to erode trust and cooperation.
This model demonstrates powerfully how asymmetric information isn't just an abstract concept. In a vee, the structure of communication channels directly creates scenarios of incomplete information, leading to predictable and often detrimental strategic consequences. Partners who are excluded from discussions rationally anticipate the possibility of collusion and may preemptively withdraw investment in the relationship.
Conclusion
Today we've laid the crucial groundwork for analyzing relationships where information is not equally shared. We've moved beyond simple games and into the more realistic domain of Bayesian games.
Key Takeaways:
- Incomplete Information refers to uncertainty about players' payoffs or "types," a common feature in all relationships.
- We model this by introducing "Nature" to choose player types and by defining players' beliefs (probabilities) about the types of others. This converts the game into one of complete but imperfect information.
- The solution concept for these games is the Bayesian Nash Equilibrium (BNE), where each type of player chooses a strategy that maximizes their expected utility, given their beliefs and the strategies of other player types.
- Asymmetric communication channels are a natural and powerful way to model private information in a vee. As shown experimentally, these structures can foster collusion between the "informed" partners and lead to a rational withdrawal of cooperation from the "uninformed" partner.
We have now seen how private information can be modeled and the strategic problems it can create. But partners are not passive. They actively try to manage and overcome information gaps. In our next lesson, we will explore signaling games, where informed players take costly actions to credibly communicate their private information to others.