The Expected Utility Model described in [1] [7] computes the expected utility of actor i against actor j as the sum of expected utilities in two different scenarios: when i challenges j and when it does not. In the challenging case, the expected utility includes two possible outcomes, depending on whether j decides to challenge back. The probability that j does not challenge i is represented by ( 1 − s j ) , while s j represents the probability of a challenge. The utility gained in case of success and failure for actor i is denoted as U s i i and U f i i , respectively, in (2).

When i is not challenging j, the expected utility includes two different scenarios as well, which are represented in (3). In the first scenario, actor i goes through the status quo with a probability of Q, where Q is either 0.5 or 1, depending on the setting. The utility gained in the status quo is U s q i . In the second scenario, actor i does not go through the status quo with a probability of ( 1 − Q ) . In this scenario, actor i can experience a better or worse situation with a probability of T or ( 1 − T ) , respectively. T is set to 1 when d i s t ( i , μ ) < d i s t ( i , j ) , where μ is the median voter position, and 0 otherwise.

The structure of the model is visualized as a tree in Figure 1. The computation of the expected utility is shown in (1), where ( E U i j i ) c and ( E U i j i ) n c represent the expected utilities in the challenging and non-challenging cases, respectively.

It is worth noting that the definition of basic utilities U s i i , U f i i , U s q i , U b i i , U w i i for a two-dimensional model are provided in the next section. For more details on the one-dimensional model, the reader can refer to [7] [9] .

E U i j i = ( E U i j i ) c − ( E U i j i ) n c (1)

( E U i j i ) c = s j ( p i j i U s i i + ( 1 − p i j i ) U f i i ) + ( 1 − s j ) U s i i (2)

( E U i j i ) n c = Q U s q i + ( 1 − Q ) ( T U b i i + ( 1 − T ) U w i i ) (3)

The basic utilities U s i i , U f i i , U s q i , U b i i , U w i i are defined as follow. For more details on the formulation please see [1] [6] [7] .

If actor i is contemplating a new position that differs from its current position, its preferred outcome, then its utility function can be defined as a decreasing function of the distance between the two positions. Hence, the utility function is highest for actor i when the alternative position is the same as its current position and is lowest when the two positions are at opposite ends of the position range, as viewed by actor i. The equation for this function is given below:

u i j i = f ( − | x i − x j | ) (4)

where x i is actor i’s position and f is any arbitrary descending function. The utility function u i j i demonstrates how much actor i attaches to his own policy portfolio. The specific function f that is used in our model is:

f ( σ ) = 1 − 2 σ r i (5)

where the risk parameter, denoted by r i , ranges from 0.5 to 2 and will be defined mathematically in a subsequent section. This parameter is used to assess an actor’s willingness to take risks in the decision-making process. Players who hold positions with less support from other players are considered more risk-seeking, ideological, or ambitious, and are likely to have lower values of r i (minimum value of 0.5). Conversely, players who are risk-averse tend to hold positions that are closer to the current likely outcome and do not have to engage in conflicts with other players to reach an agreement and are likely to have higher values of r i (maximum value of 2) [8] . Substituting (5) into (4), the utility function would become:

u i j i = 1 − 2 | x i − x j | r i (6)

where u i j i ∈ [ − 1,1 ] and x i and x j are normalized so that x i , x j ∈ [ 0,1 ] . The figure depicted in Figure 2 illustrates the alterations in the utility function of an actor as it moves away from its intended position, assumin that the actor’s risk is given by r i = 1 . The application of the risk parameter is reflected in the utility functions of stakeholders, as shown in Figure 3. The introduction of risk preferences in stakeholder behavior leads to changes in their utility assessments. Stakeholders who are more willing to take risks exhibit utility functions with a convex shape, resulting in a rapid decline in their utility as they move further away from their preferred position. In contrast, stakeholders who are risk-averse exhibit utility functions with a concave shape, where their utility decreases slowly as they move away from their ideal position.

In a voting process to determine the median voter position, each actor’s level of support for their preferred position must be assessed. Actor i casts a vote between positions j and k, indicating their preference for one over the other. The vote is calculated as follows:

v j k i = c i s i ( u i j i − u i k i ) (7)

where u i j i is the utility function of actor i for challenging actor j from i’s point of view, c i and s i are actor i’s capability and salience, respectively, and v j k i is the resulting vote. By replacing (6) into (7), we can obtain the following expression:

v j k i = 2 c i s i ( | x i − x k | − | x i − x j | ) (8)

The Condorcet method of voting involves comparing the preference level of one position over another in a pair-wise voting system to determine which position receives the most votes. In this method, the position that receives the most votes is declared the winner. The preference level achieved by each position is determined by the vote cast by each actor, as described in the previous step. This process helps to identify the position that has the most support among the actors and is therefore most likely to be accepted as the median voter position. It is important to note that the median voter position may not necessarily be the optimal solution for the problem at hand, but rather a compromise that satisfies the preferences of the majority of actors. Using the preference level achieved by position j compared to position k, the median voter position μ can be obtained by finding the position with the most support. This calculation can be performed using the following equation:

v j k = ∑ i = 1 , i ≠ j , k n     v j k i (9)

where v j k is the votes cast for j versus k from all other players’ point of view.

When two actors i and j challenge each other, the probability of success p i j can be expressed as the ratio of the total support received by one actor over the other:

p i j = ∑ k | u k i > u k j v i j k ∑ k = 1 n | v i j k | (10)

The probability of success, p i j , can be interpreted as the amount of support received by actor i in comparison to actor j [14] . By substituting (8) into (10), the i’s probability of success over j can be achieved by:

p i j = ∑ k | a r g > 0 c k s k ( | x k − x j | − | x k − x i | ) ∑ k = 1 n c k s k | ( | x k − x j | − | x k − x i | ) | (11)

where c k and s k are the capability and salience of player k in the issue. The numerator of (11) sums up the preference votes of all actors who prefer actor i over actor j. The denominator sums up the support for both actors, resulting in a probability value between 0 and 1.

The risk value is used to measure an actor’s willingness to take risks in order to diverge from the median voter position. Actors with desired positions closer to the median are less likely to require intense negotiations and are therefore less likely to take risks, resulting in a risk-averse strategy. Conversely, actors with weaker support and weaker alliances are more likely to take risks, resulting in a risk-seeking strategy. The risk term for actor i, denoted as r i involves the calculation of risk value R i given below:

R i = 2 ∑ j = 1 , j ≠ i n     E U j i i − max i ∑ j = 1 , j ≠ i n     E U j i i − min i ∑ j = 1 , j ≠ i n     E U j i i max i ∑ j = 1 , j ≠ i n     E U j i i − min i ∑ j = 1 , j ≠ i n     E U j i i (12)

where R i ∈ [ − 1,1 ] . Once R i is obtained, the risk term r i can be computed using a transformation in (13) which guarantees 0.5 < r i < 2 . Initially, all actors are considered to be risk neutral and assigned a risk term of 1, and it is recalculated for each round.

r i = 1 − R i / 3 1 + R i / 3 (13)

In this section, we introduce the basic utilities consisting of U s i i , U f i i , U s q i , U b i i , and U w i i . Given the assumption that actor i either defeats or wins over actor j, the corresponding utilities of success U s i and failure U f i are derived by actor i as follows:

U s i i = 2 − 4 [ 0.5 − 0.5 | x i − x j | ] r i (14)

U f i i = 2 − 4 [ 0.5 + 0.5 | x i − x j | ] r i (15)

where 2 − 4 ( 0.5 ) r i ≤ U s i i ≤ 2 and − 2 ≤ U f i i ≤ 2 − 4 ( 0.5 ) r i . Referring to Figure 1, it can be observed that if actor i chooses not to challenge actor j, actor j may either maintain their current position as status quo or move, which may result in a better or worse situation for actor i, and can be associated with the utilities U b i and U w i , respectively. Therefore, the utilities U b i i , U w i i , and U s q i are defined as follows to satisfy the condition U w i i ≤ U q i i ≤ U b i i :

U b i i = 2 − 4 [ 0.5 − 0.25 ( | x i − μ | + | x i − x j | ) ] r i (16)

U w i i = 2 − 4 [ 0.5 + 0.25 ( | x i − μ | + | x i − x j | ) ] r i (17)

where 2 − 4 ( 0.5 ) r i ≤ U b i i ≤ 2 and − 2 ≤ U w i i ≤ 2 − 4 ( 0.5 ) r i , and μ is the median voter position which can be calculated using (9) to find the position with the most support. In a case where i does not challenge j and j does not move, the status quo utility is realized and is defined as:

U s q i = 2 − 4 [ 4 − 0 8 ] r i = 2 − 4 ( 0.5 ) r i (18)

The value of parameter Q in (3) has been set to different values (0.5 or 1) in various research papers. For instance, in [15] , Q is considered as 1, while in [16] , it is taken as 0.5. However, in our study, we have assumed Q to be 1. On the other hand, parameter T determines whether the situation gets better for actor i or not. If d i s t ( i , μ ) < d i s t ( i , j ) , then T is set to 1, indicating that the situation has improved for actor i. Otherwise, if d i s t ( i , μ ) ≥ d i s t ( i , j ) , then T is set to 0.

The expected utilities are utilized to determine the negotiation outcomes among actors, which can fall into one of four categories: conflict, capitulation, compromise, or stalemate. Figure 4 illustrates these scenarios and the corresponding expected utilities for each actor. Further details on each scenario are defined below.

• Conflict: When actors i and j both believe that they have the upper hand, a conflict is likely to occur, and the outcome of the conflict is determined based on the following conditions:

- If E U i j i > E U j i j , actor j moves to i’s position.

- If E U i j i < E U j i j , actor i moves to j’s position.

- Actor j moves to i’s position when E U i j i > E U j i j .

- Actor i moves to j’s position when E U i j i < E U j i j .

• Capitulate: If E U i j i > 0 , E U j i j < 0 , and | E U i j i | < | E U j i j | , actor j is expected to capitulate and accept actor i’s current position.

p r o p o s a l i j i = x i (19)

• Compromise: If actor i has the upper hand and actor j is willing to agree to an acceptable offer, they compromise in favor of actor i, and the offer is closer to i’s position. This scenario occurs when E U i j i > 0 , E U j i j < 0 , and | E U i j i | > | E U j i j | .

Δ x = ( x i − x j ) | E U i j j E U j i i | (20)

p r o p o s a l i j i = x j + Δ x (21)

• Stalemate: If both actors believe that they cannot beat the other, then they do not move from their current positions, resulting in a stalemate. This scenario happens when E U i j i < 0 , E U j i j < 0 .

At the end of each round, actors receive offers from others, and the question is which offer to accept to determine each actor’s position for the next round. Mesquita [17] and Baranick [18] suggested that the actor should choose the offer that requires the smallest possible movement. We implemented this idea and assumed that the minimum move should be greater than zero unless all received offers are the same as the actor’s current position. Future studies may investigate for a more efficient offer selection algorithm.

One way to realize this idea is to consider each actor’s position function as a Gaussian distribution. This distribution has a bell-shaped curve and can be skewed towards either direction. It is bounded within the interval [0, 1] and can be truncated at any point within this range. The parameters of the distribution, namely the mean, variance, and truncation points, can be estimated from the data or set based on prior knowledge.

In order to incorporate Gaussian distribution into the existing dataset, we can assign the initial position as the mean of the distribution and the variance can be set to be proportional to the inverse of salience. The mathematical formulation of the Gaussian distribution is presented below:

p i = p i ( x ) = 1 σ 2 π exp ( − ( x − m x ) 2 2 σ 2 ) (22)

where m x represents the mean value and σ 2 represents the variance of the Gaussian distribution.

The position distribution of actors is illustrated in Figure 5 using Gaussian functions with varying means and variances. The figure reveals that actors with higher salience, and thus lower variance, are less likely to share a common position with other actors.

By applying lower and upper bounds to the variable in the normal distribution, we can obtain the truncated normal distribution. The mathematical representation of the truncated normal distribution is as follows:

f ( x ; μ , σ , A , B ) = 1 σ ϕ ( x − μ σ ) Φ ( B − μ σ ) − Φ ( A − μ σ ) (23)

ϕ ( x ) = 1 2 π exp ( − 1 2 x 2 ) (24)

Φ ( x ) = 1 2 ( 1 + erf ( x / 2 ) ) (25)

where μ and σ represent the mean and standard deviation of the normal distribution, respectively, and A and B are the lower and upper bounds of the truncated distribution. ϕ ( . ) and Φ ( . ) denote the probability density function and cumulative distribution function of the standard normal distribution, respectively.

By setting A = − ∞ , we get Φ ( A − μ σ ) = 0 . Similarly, if B = + ∞ , we obtain Φ ( B − μ σ ) = 1 . Thus, we can use the truncated normal distribution to represent one-sided position functions.

Figure 6 shows an example of the truncated normal distribution with different lower and upper bounds. We can observe that the probability density function of the truncated normal distribution is zero outside the given bounds.

Another family of distributions that can be used for modeling position functions is the beta distribution. This distribution can be bounded within the interval [0, 1], and its shape is flexible, allowing it to be either skewed or symmetric. The beta distribution has two shape parameters that control the skewness and kurtosis of the distribution, and these parameters can be estimated using maximum likelihood or Bayesian methods.

The beta family of distributions can be defined over a general interval (A, B) by the following equation:

f ( x ; α , β , A , B ) = 1 B − A Γ ( α + β ) Γ ( α ) ⋅ Γ ( β ) ( x − A B − A ) α − 1 ( B − x B − A ) β − 1 (26)

where α and β are the shape parameters, and A and B denote the interval endpoints. However, if the interval is confined to [0, 1], using A = 0 and B = 1 in (26) yields the standard Beta distribution:

f ( x ; α , β ) = ( Γ ( α + β ) Γ ( α ) ⋅ Γ ( β ) x α − 1 ( 1 − x ) β − 1 0 ≤ x ≤ 1 0 otherwise (27)

where μ and σ can be calculated as:

μ = α α + β (28)

σ 2 = α β ( α + β ) 2 ( α + β + 1 ) (29)

An alternative formulation of the beta distribution involves reparameterizing it in terms of μ = α α + β and ϕ = α + β [23] . The distribution can then be expressed as:

f ( x ; μ , ϕ ) = Γ ( ϕ ) Γ ( μ ϕ ) ⋅ Γ ( ( 1 − μ ) ϕ ) x μ ϕ − 1 ( 1 − x ) ( 1 − μ ) ϕ − 1 ,     0 ≤ x ≤ 1 (30)

where 0 < μ < 1 and ϕ > 0 . Using this formulation, μ can be used as the mean parameter, and ϕ can be interpreted as the precision parameter. Greater precision yields a smaller variance if μ is fixed. Figure 7 illustrates the beta distribution for various combinations of μ and ϕ . The figure presents the probability density function of the standard beta distribution for different combinations of its shape parameters, μ and ϕ . The beta distribution is a flexible distribution that can model a variety of data types and is particularly useful for modeling data that are bounded between 0 and 1. In subplot (a), we have fixed ϕ = 5 and varied μ from 0.05 to 0.95 in increments of 0.25. In subplots (b), (c), and (d), we have the same setup, but with ϕ fixed at 15, 50, and 100, respectively. As μ increases, the peak of the distribution shifts towards the right and the tails become thinner. As ϕ increases, the distribution becomes more concentrated around the mean and the tails become even thinner. The figure provides insights into

how the beta distribution changes with varying values of its shape parameters and can aid in selecting appropriate parameter values for modeling a position for a given actor. The utilization of beta distribution as a position function allows for the interpretation of the highest point of the distribution as the desired position, while the degree of sharpness in its tail can be regarded as an indication of the level of significance (salience) of the issue to the actor.

In addition to the beta distribution, the triangular distribution is another candidate for modeling actors’ position function. This distribution is particularly useful when prior knowledge about the range and mode of the position function is available. The triangular distribution can be bounded within the interval [0, 1] and is characterized by three parameters: the lower and upper bounds, and the mode. The skewness of the position can be adjusted by controlling the values of the lower and upper bounds.

The triangular distribution is a continuous probability distribution over the interval [a, b] and can be defined as follows:

f ( x ; a , b , c ) = ( 0     x < a , 2 ( x − a ) ( b − a ) ( c − a ) a ≤ x < c , 2 b − a     x = c , 2 ( b − x ) ( b − a ) ( b − c ) c < x ≤ b , 0     b < x . (31)

where a ∈ ( − ∞ , ∞ ) is the lower limit, b > a is the upper limit, and a ≤ c ≤ b . Parameter c is the mode of the distribution and can be considered as the central position taken by the actor. To utilize this distribution as a position function limited within [0, 1], it is necessary to select the values of parameters a and b from the interval a ∈ [ − 1 , 1 ] for our specific purposes. The skewness of the position can be set by parameters a and b. Figure 8 displays the triangular distribution for various combinations of parameters a, b, and c over the interval [0, 1].

Using either the beta or the triangular distribution as a position function, the peak of the distribution can be considered as the desired position, and the thinness of its tail can represent the level of salience of the issue to the actor. The choice of which distribution to use depends on the availability of prior knowledge about the position function and the desired level of control over the skewness of the distribution. Figure 8 illustrates the flexibility of the triangular distribution in this regard. We can observe that as the difference between the mode and the midpoint of the interval ( a + b ) / 2 increases, the distribution becomes more skewed.

Various techniques can be employed to quantify the distance between two stochastic variables and can be applied in our model to evaluate the distance between position functions. The choice of distance metric can depend on the nature of the data and the research question being addressed. In the following, we will examine a few of them.

By incorporating the Lukaszyk-Karmowsky distance function into the BDM model and treating the discrete position representation as a position function with a Dirac delta function, the same results can be achieved. The position function for actor i can be defined as:

p i = p i ( x ) = δ ( x − x i ) (37)

where x 0 is the initial position of actor i. Figure 9 represents two actors in the BDM position model using Dirac functions. Previously, the distance between the positions of two actors i and j was obtained using:

D ( x i , x j ) = | x i − x j | (38)

However, using (32) to calculate the distance between two positions, we get the same result:

D ( X , Y ) = ∫ − ∞ ∞     ∫ − ∞ ∞ | x − y | δ ( x − x 0 ) δ ( y − y 0 ) d x d y = ∫ − ∞ ∞ | y − x 0 | δ ( y − y 0 ) d y = | y 0 − x 0 | (39)

where x 0 and y 0 are actors i and j’s initial positions (discrete positions), respectively.

The median voter position with the maximum support can be calculated using (9). However, to calculate U b i i and U w i i , the distance between the median voter position ( μ 0 ) and the distributed position function, p i , needs to be determined. In order to calculate this distance, we can consider a position function for the median voter position as a truncated normal distribution with standard deviation of σ = 0.5 and average of μ = μ 0 . The distance between the actor’s position and the median voter position can then be calculated using a distance measure, such as the Lukaszyk-Karmowsky Distance Function.

In dealing with problems that involve large number of issues (dimensions) and/or large number of actors per issue/dimension, the computational complexity of the problem becomes increasingly important. To demonstrate the analysis of the model’s time complexity, a simple pseudo code is provided in Algorithm 10. Table 1 displays the complexity of different parts of the algorithm. The overall computational complexity of the algorithm is O ( M 3 ) , where M is the number of actors. The main calculation in the for loops involves calculating the distance functions using the Lukaszyk-Karmowsky Distance Function. This involves calculating dual integral to find the distance between two position functions. To reduce computational time, we can pre-calculate the pairwise distance between all actors at the beginning of each round and store them in a lookup table for later retrieval. This optimization can reduce the time complexity remarkably, which can now be estimated as O ( M 2 ) . However, the space complexity remains unchanged at O ( M 2 ) .

In this subsection, we compare four different distance functions to find the most suitable one for measuring the distance between truncated normal distributions as position functions in our model. The four distance measures we considered are the Lukaszyk-Karmowsky Distance Function, the Bhattacharyya Distance Function, the Cosine Similarity-based Distance Function, and the Normalized Similarity Measure based Distance Function. It is worth noting that the suitability of each distance function may vary depending on the specific characteristics of the distributions being compared. Therefore, it is recommended to perform a comprehensive comparison of different distance functions for any given problem to select the most appropriate one.

To conduct the comparison, we generated four scenarios of truncated normal distributions with different parameters. These scenarios can be seen in Figure 10. For each subplot, we calculated the distance between the two truncated normal distributions using each of the four distance functions. The results of our comparison are summarized in Table 2 below. The first row in the table shows the parameters of the truncated normal distributions used depicted in Figure 10 subplots, where (0.2, 0.1) vs (0.3, 0.1) represents the comparison between the truncated normal distributions in subplot (a) with means 0.2 and 0.3, and standard deviations 0.1, respectively.

In our proposed model, we hypothesized that the degree of overlap between position functions would inversely affect their distance, i.e., the distance would be higher when there is no overlap and lower when the functions share a common area. Additionally, when comparing two-position functions with similar μ but low σ (Figure 10(b) and Figure 10(d)), we expect the distance to approximate the difference between their mean values. Our analysis using Table 1 reveals that only the Lukaszyk-Karmowsky distance function satisfies this expectation, and we have thus selected it for our model. However, it is important to note that for some case studies with similar positions like those in Figure 10(b) and Figure 10(d), maximum distance may be preferred, and for such situations, the cosine similarity or normalized similarity measure based distance function could be considered.

The EMU Positions dataset is a collection of data related to the major reforms of the Economic and Monetary Union that took place between 2010 and 2015. It covers 47 contested policy issues that were negotiated between 28 EU member states and significant EU institutions, including their position and salience. The dataset was created using various data collection methods, including expert interviews and text analysis, in order to improve the accuracy and comprehensiveness of the data [13] .

The dataset contains information about the positions taken by different actors during the negotiation process, as well as the salience of these positions. In the online appendix and the codebook, they provide the data and detailed information on it. By analyzing this data, researchers can gain insights into the decision-making process of the EU legislative body, as well as the factors that influence decision-making.

It is important to note that the dataset had missing position or salience information for some actors in certain issues. Therefore, those actors were eliminated from the analysis of those issues. To estimate the actors’ capability in the issues, their GDP for the year 2021 was used as a proxy for their level of power. This assumption is considered reasonable given that the issues being analyzed are related to financial and monetary policies. Although the actors’ GDP may have been different at the time the dataset was created, we assume that their relative power remains almost the same. Table 3 shows the capability values used for the actors present in the issues.

To evaluate the model’s capability in predicting negotiation problems, we ran the model on the EMU position dataset and compared its performance with the BDM model and BiLSTM model. In our model, we used a truncated normal distribution with μ = x i and σ proportional to 1 / s i as the position function for each actor. Equation (40) was used to calculate the sigma value used in the position function. We considered p s = 1 , m s = 100 , and ε = 0.01 as the parameters.

σ = 1 m s ( s i ) p s + ε (40)

where s i is actor i’s salience. The parameter ε is used as denominator offset to avoid division by zero, and p s and m s are reserved as degrees of freedom in the implementation of the position function. For this experiment, the BiLSTM architecture had 10 neurons in the first layer and 5 neurons in the second layer.

The evaluation metric used was the mean absolute error (MAE). Table 4 shows the comparison of MAE for the proposed model, BDM model, and the BiLSTM model. Our position function based model outperformed both the BiLSTM model and BDM model with an MAE of 0.2122, indicating its effectiveness in predicting negotiation outcomes.

To present the explainability of the proposed model, we employed a one-dimensional case study from [5] . The issue is to predict the number of years it would take for the introduction of emission standards for medium-sized automobiles. Table 5 shows the initial capabilities, positions, and salience of the players involved in the problem. Figure 11 depicts the positions of the players after 10 rounds. The results obtained are in agreement with the findings reported in [6] . According to [6] , the expected utility model predicted the outcome to be 7.0 years, while the actual delay was 8.83 years. According to the median voter position result, our model predicts the outcome to be 7.0 years. From Figure 11, it can be seen that the actors are negotiating during each round until they reach an agreement in the final round. For instance, the UK is gradually changing its position, while Italy is changing its position at a faster pace. Although the capability of the UK and Italy is considered equal towards the issue, it can be inferred that the UK has more salience towards the issue, which is why it changed its position at a slower pace than Italy. Another instance could be a comparison between Denmark and Germany. Since Denmark’s capability is lower than Germany’s, it left its initial position earlier.