# Can Social Agents Efficiently Perform in Automated Negotiation?
## 3 Proposed Social Agent
### Opponent Modeling
#### 3.5.2 Issue Weight Estimation  
The proposed model seeks statistically significant changes in issue value distributions of the two consecutive, disjoint and time-sorted window of offers as shown in Figure 3. The underlying assumption is that, when a concession is carried out, the distribution of issue values changes from window to window. Thus, if one detects changes between in the distribution of issue values sent by the opponent, one may detect concession moves. Before going further, we explain the concept of issue value distribution in more detail. Equation (6) formulates the issue value distributions in a list of offers \(\mathcal{O}\). Given a window of bids, the formula calculates the occurrence frequencies of each of the issue \(i\)'s negotiation values \(j\), \(Fr_{i}(j,\mathcal{O})\). The \(\delta_{i}\) returns 1 if the value \(j\) of issue \(i\) exists in the offer \(o\). The \(\mid\mathcal{O}\mid\) denotes the length of the bid window. The formula applies Laplace smoothing to handle the erroneous outcomes of the statistical significance test in the Algorithm 2. Table 1 demonstrates how this calculation is performed for a given example. \[Fr_{i}(j,\mathcal{O})=\frac{1+\sum_{o\in\mathcal{O}}\delta_{i}(j,o)}{n+\mid \mathcal{O}\mid} \tag{6}\]  
```
Data:\(t\): The current time in the negotiation, \(\mathcal{O}^{\prime}\): The previous partition of \(k\) offers, \(\mathcal{O}\): The current partition of \(k\) offers, \(\mathcal{O}_{1\to t}\): All offers received so far, \(\mathcal{W}^{\prime}=\{w^{\prime}_{1},\dots,w^{\prime}_{n}\}\): The current weights for the opponent model Result:\(\mathcal{W}=\{w_{1},\dots,w_{n}\}\): The new weights for the opponent model \(e\gets 2\); \(\text{concession}\leftarrow\text{False}\); foreach\(i\in\mathcal{I}\)do \(\vec{F}^{\prime}_{i}\leftarrow(Fr_{i}(1,\mathcal{O}^{\prime}),\dots,Fr_{i}(n, \mathcal{O}^{\prime}))\); \(\vec{F}_{i}\leftarrow(Fr_{i}(1,\mathcal{O}),\dots,Fr_{i}(n,\mathcal{O}))\); \(p_{val}\leftarrow\mathcal{X}^{2}\text{-test}(\vec{F}_{i}=\vec{F}^{\prime}_{i})\); if\(p_{val}>0.05\)then \(e\gets e\cup\{i\}\); else \(\vec{V}_{i}\leftarrow(\hat{V}_{i}(1),\dots,\hat{V}_{i}(n))\); \(E[\mathcal{U}_{i}(\mathcal{O}^{\prime})]\leftarrow\vec{V}_{i}\times\vec{F}^{ \prime}_{i}\); \(E[\mathcal{U}_{i}(\mathcal{O})]\leftarrow\vec{V}_{i}\times\vec{F}^{\prime}_{i}\); if\(E[\mathcal{U}_{i}(\mathcal{O})]<E[\mathcal{U}_{i}(\mathcal{O}^{\prime})]\)then \(\text{concession}\leftarrow\text{True}\);  end if  end for  end for if \(|e|\
eq n\)and concession\(=\text{True}\)then foreach\(i\in e\)do \(w_{i}\gets w^{\prime}_{i}+\Delta(t)\)  end if else \(W\gets W^{\prime}\)  end for
```  
**Algorithm 2**The issue weight update mechanism  
``` Data:\(t\): The current time in the negotiation, \(\mathcal{O}^{\prime}\): The previous partition of \(k\) offers, \(\mathcal{O}\): The current partition of \(k\) offers, \(\mathcal{O}_{1\to t}\): All offers received so far, \(\mathcal{W}^{\prime}=\{w^{\prime}_



The proposed algorithm departs from classic frequentist opponent modeling approaches in two ways: it employs non-sliding windows rather than two consecutive offers to decide how to update weights, and it is based on statistical tests of significance to compare the issue value distribution in windows of bids. The proposed algorithm iterates over every single negotiation issue \(i\) and calculates the frequency distribution of the issue values in the previous window \(\vec{\mathcal{F}}_{i}^{t}\) and the frequency distribution of the issue values in the current window \(\vec{\mathcal{F}}_{i}\) (lines 3-5). Then, a Chi-squared test is carried out with the null hypothesis being that both frequency distributions, \(\vec{\mathcal{F}}_{i}\) and \(\vec{\mathcal{F}}_{i}^{t}\), are statistically equivalent (line 6). The main goal behind this test is to check whether the distribution of issue values for \(i\) changed from the previous window of offers to the current one. This information helps us determine if, overall, the opponent changed the type of offers sent.