Building Bayesian Networks for Negotiation Dynamics

Why Bayesian Networks for Negotiation?

At its heart, negotiation is a process of learning. Each proposal, counter-proposal, and conversation provides signals that allow you to update your beliefs about your counterpart's preferences, limits, and even their "type." A formal model of this process is essential for strategic thinking.

As the paper highlights, a negotiator's actions are based on their beliefs about the world and their counterparts. As they receive new information—an offer, a reaction, a piece of external news—they should update these beliefs. The Bayesian framework provides the mathematical engine for this updating process.

The Building Blocks of a Bayesian Network

A Bayesian Network is a directed acyclic graph (DAG) where nodes represent random variables and directed edges represent conditional dependencies. It's a compact way to specify a full joint probability distribution over all the variables.

The "Alarm" network illustrates a key insight: observing one cause (Earthquake) for an effect (Alarm) "explains away" the need for another cause (Burglary), reducing its probability. This pattern of inter-causal reasoning is fundamental and appears frequently in complex diagnostic and predictive tasks.

Formally, a Bayesian Network consists of:

1. A Network Structure (S): A DAG where each node corresponds to a random variable $X_i$. The graph structure encodes a set of conditional independence assumptions.
2. A set of Parameters (P): For each variable $X_i$, there is a Conditional Probability Distribution (CPD), often represented as a Conditional Probability Table (CPT), $P(X_i|\text{Parents}(X_i))$.

Together, these components define the joint probability distribution over all variables as the product of the local conditional distributions:

Constructing a Bayesian Network for Partner Selection

Let's build a BN for a scenario relevant to our course. Imagine Wendy is considering a first date with a potential new partner, Brian. She and her husband, Charles, want to model the situation to better understand the risks and potential outcomes. Their goal is to infer Brian's underlying "type" and predict the date's success based on early signals.

Here is a possible BN for this scenario.

1. Define Variables:

• Partner Type (T): Brian's underlying character concerning the lifestyle. This is a hidden variable. States: {Respectful, User, Clueless}.
• Communication Style (C): Brian's observed communication during initial chats. States: {Open, Guarded}.
• Boundary Response (B): Brian's reaction when Wendy mentions a key boundary (e.g., "I don't play on the first date"). States: {Accepts, Pushes}.
• Date Success (S): Whether the date is a success from Wendy's point of view. States: {Yes, No}.
• Relationship Impact (I): The impact on Wendy and Charles's primary relationship. States: {Positive, Neutral, Negative}.

2. Define Network Structure (The Graph):
We draw directed edges from causes to effects.

• A partner's Type influences their Communication Style and their Boundary Response. So, T -> C and T -> B.
• Both Communication Style and Boundary Response influence Date Success. So, C -> S and B -> S.
• Date Success, in turn, affects the Relationship Impact. So, S -> I.

This gives us the following structure:

graph TD
    T(Type) --> C(Communication)
    T --> B(Boundary Response)
    C --> S(Date Success)
    B --> S
    S --> I(Relationship Impact)

3. Define Parameters (The CPTs):
Now, we must specify a conditional probability table for each node, given its parents. These would be based on Charles and Wendy's prior beliefs and experience.

• P(T): Their prior belief about men in the dating pool.

  T	P(T)
  Respectful	0.5
  User	0.3
  Clueless	0.2

• P(C | T): How likely is a communication style given the man's type?

  T	C=Open	C=Guarded
  Respectful	0.8	0.2
  User	0.3	0.7
  Clueless	0.6	0.4

• P(B | T): How likely is he to push boundaries given his type?

  T	B=Accepts	B=Pushes
  Respectful	0.9	0.1
  User	0.2	0.8
  Clueless	0.7	0.3

• P(S | C, B): How do communication and boundaries affect date success?

  C	B	S=Yes	S=No
  Open	Accepts	0.95	0.05
  Open	Pushes	0.30	0.70
  Guarded	Accepts	0.50	0.50
  Guarded	Pushes	0.05	0.95

• P(I | S): How does a successful date impact the primary relationship?

  S	I=Positive	I=Neutral	I=Negative
  Yes	0.8	0.2	0.0
  No	0.0	0.3	0.7

This process of defining variables, determining causal structure, and assessing local probabilities is the standard method for building a BN from expert knowledge.

Reasoning with the Network: Probabilistic Inference

With the BN defined, we can perform inference—that is, ask it questions.

Query 1: Prediction
Before any interaction with Brian, what is the probability of a successful date, P(S = Yes)?
To answer this, we would sum the joint probability over all possible states of the other variables, a process called marginalization.
$P(S)={\sum }_{T,C,B,I}P(T,C,B,S,I)$
Using the network structure, this simplifies to:
$P(S)={\sum }_{T,C,B}P(S|C,B)P(C|T)P(B|T)P(T)$
Plugging in the numbers from our CPTs would give a prior probability of success.

Query 2: Diagnosis/Belief Updating
Wendy chats with Brian. His communication seems Guarded (C=Guarded), and when she mentions a boundary, he Pushes back (B=Pushes). What is the new probability that he is a User, i.e., P(T=User | C=Guarded, B=Pushes)?

This is where the power of Bayesian reasoning comes in. We use Bayes' rule:
$P(T|C,B)=\frac{P(C,B|T)P(T)}{P(C,B)}$
Thanks to the conditional independence assumptions in our network (C and B are independent given T), this simplifies to:
$P(T|C,B)\propto P(C|T)P(B|T)P(T)$
The term $\propto$ means "proportional to," acknowledging that we'd need to normalize the result. Let's calculate the unnormalized posterior belief for each type:

• Respectful: $P(C_G|T_R)P(B_P|T_R)P(T_R)=0.2\times 0.1\times 0.5=0.01$
• User: $P(C_G|T_U)P(B_P|T_U)P(T_U)=0.7\times 0.8\times 0.3=0.168$
• Clueless: $P(C_G|T_C)P(B_P|T_C)P(T_C)=0.4\times 0.3\times 0.2=0.024$

After normalizing (dividing each by the sum 0.01 + 0.168 + 0.024 = 0.202), we get:

• P(T=Respectful | evidence) = 0.01 / 0.202 ≈ 5%
• P(T=User | evidence) = 0.168 / 0.202 ≈ 83%
• P(T=Clueless | evidence) = 0.024 / 0.202 ≈ 12%

Our initial belief that Brian was a User was 30%. After observing just two signals, our belief has jumped to 83%. This is a quantitative example of "red flags" in action and demonstrates how a BN can formalize this intuitive reasoning process.

Learning a Bayesian Network from Data

We built our network from "expert knowledge." But what if we have a dataset of past first dates? We can learn the network's parameters (and even its structure) from this data.

Learning Parameters for a Known Structure: If we assume our graph structure is correct, learning the CPTs from a complete dataset is straightforward. As you know from your statistics background, it reduces to calculating the conditional frequencies in the data. For example, to learn $P(S=\text{Yes}|C=\text{Open},B=\text{Accepts})$, we would just count the fraction of times a date was successful among all dates in our dataset where communication was open and boundaries were accepted. This is a Maximum Likelihood Estimation (MLE) of the parameters.

Learning with Incomplete Data / Hidden Variables: In our example, the Partner Type (T) is a hidden variable; we never observe it directly. This makes learning much more interesting and challenging. We can't just count. This is where algorithms like Expectation-Maximization (EM) come in, which you may have encountered. The EM algorithm iteratively "fills in" the missing values based on the current best guess of the parameters and then re-estimates the parameters based on the filled-in data.

Conclusion

In this lesson, we've explored Bayesian Networks as a formal system for representing and reasoning about uncertain knowledge. You've seen how a BN's structure captures causal assumptions and how its parameters quantify the strength of those relationships. Most importantly, you've seen how to use a BN to perform inference—updating your beliefs about hidden variables like a potential partner's "type" in light of observable evidence like their communication style.

Key Takeaways:

• A Bayesian Network is a Directed Acyclic Graph that provides a compact representation of a joint probability distribution.
• The structure is built by drawing arrows from causes to effects, encoding conditional independence assumptions.
• The parameters (CPTs) quantify the probabilistic relationships between a node and its parents.
• Inference is the process of querying the network, most powerfully used to update beliefs about hidden variables given new evidence (e.g., P(Cause | Effect)).
• The phenomenon of "explaining away" is a key reasoning pattern where confirming one cause of an effect makes other potential causes less likely.
• The parameters of a BN can be learned from data, using simple counting for complete data or algorithms like EM for data with hidden variables.

Preview of the Next Lesson:
This was the final new tool in our modeling toolkit. In our next lesson, we will synthesize the concepts from the entire course—game theory, bargaining models, simulations, and Bayesian networks—to develop a comprehensive case study analyzing a fictional relationship dynamic from start to finish. This will be your opportunity to see how these different models fit together to provide a multi-faceted analysis of strategic interaction in relationships.