Title: Preference Banzhaf: A Game-Theoretic Index with Feature-wise Probabilities
Venue: NeurIPS 2025
Accepted: no
Keywords: XAI

Game Theory

Banzhaf Value

Feature Attribution
Abstract: Game-theoretic feature attribution methods are popular in XAI because they satisfy several desirable axioms. Approximating a model as a game with input features as players, these methods measure the weighted average contribution of each feature to a model's prediction across different feature subsets. However, these techniques also make strict assumptions that may affect the quality of the explanations. One common assumption is that all features can join or leave a subset with probability of 0.5, i.e., all subsets are equally likely to form. However, in real games, each player can have different preference for joining a coalition, shifting the probability of the subsets and thus the attribution values. Following this notion, we introduce Preference Banzhaf, which calculates Banzhaf-like value with adjusted probabilities using centered linear regression. We theoretically show the convergence of Preference Banzhaf and empirically demonstrate the effect of probability adjustment on explanation quality and sensitivity.

1

Abstract

Game-theoretic feature attribution methods are popular in XAI because they satisfy several desirable axioms. Approximating a model as a game with input features as players, these methods measure the weighted average contribution of each feature to a model's prediction across different feature subsets. However, these techniques also make strict assumptions that may affect the quality of the explanations. One common assumption is that all features can join or leave a subset with probability of 0.5, i.e., all subsets are equally likely to form. However, in real games, each player can have different preference for joining a coalition, shifting the probability of the subsets and thus the attribution values. Following this notion, we introduce Preference Banzhaf, which calculates Banzhaf-like value with adjusted probabilities using centered linear regression. We theoretically show the convergence of Preference Banzhaf and empirically demonstrate the effect of probability adjustment on explanation quality and sensitivity.

1 Introduction

Artificial Intelligence (AI) is becoming a ubiquitous tool in many fields thanks to their capacity 15 to reflect complicated patterns in large datasets. However, this capacity is often accompanied by 16 high model complexity, making it difficult to interpret a model's prediction process. In high-stakes 17 domains like health care or finance [1, 2], interpretability is as important as the accuracy of prediction, 18 and model complexity hinders the practical adoption of AI in these domains. Explainable AI (XAI) 19 tackles this issue by attaching explanations to the models [3, 4, 5]. 20

Among different explanations, feature attribution measures the contribution of input features to a 21

model's prediction. In particular, local model-agnostic methods compute the input importance at 22

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instance level regardless of target model's architecture [6]. There are two major branches of local

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model-agnostic attribution: Locally Interpretable Model Explanation (LIME) [7] and game-theoretic

techniques. On the one hand, LIME fits a surrogate model g θ to randomly sampled perturbations

around the target instance a linear

g

θ

since

θ

x

with a locality-defining kernel

π

.

Most LIME-based technique use corresponds directly to importance, and most improvements are derived from

modifying the noise generation process or the fitting process [8, 9, 10].

The second branch of local model-agnostic attribution is game-theoretic XAI. These methods ap29

proach the explanation process as a cooperative game, considering input features as players and the 30

model as the value function. The game theory solution of a player's contribution corresponds directly 31

to a feature's importance. The main strength of game-theoretic attribution is that they satisfy the 32

underlying axioms of the corresponding solution. For example, the Shapley value [11]: 33

is a solution of cooperative game theory that uniquely satisfies linearity, dummy, symmetry, and 34 efficiency. While the combinatorial nature of Shapley values make it impossible to calculate exactly 35 for large number of features, KernelSHAP [12] shows that it can be approximated with a weighted 36 linear regression. Due to its massive popularity, KernelSHAP has been explored thoroughly in the 37 past literature [13, 14, 15, 16]. 38

Unfortunately, KernelSHAP is suffers from issues like numerical instability. Consequently, more 39 recent literature focuses on relaxing some axioms to improve the quality of the explanations. One 40 example is the Banzhaf value [17], which is another solution of cooperative game theory which 41 satisfies the same axioms as the Shapley value except efficiency: 42

The Banzhaf value is simply an average of the payoff difference caused by player i across all possible 43 coalitions excluding said player. More generally, values of the form: 44

where p ( S ) is the probability of coalition S , are referred to as probabilistic values [18]. 45

One problem with regular Banzhaf value is that it assumes that all coalitions are equally likely to form. 46 This assumption is equivalent assuming each player being neutral to joining a coalition. However, in 47 real life, players are likely to have different preferences depending on their objectives. For example, 48 if each player wishes to maximize their payoff, a player would have a higher probability of joining 49 (i.e., a preference ) the greater their expected payoff in larger coalitions. The criteria may not even be 50 directly related to the game: for instance, if political parties vote on a regulation, they may make their 51 vote not based on the game payoff (passing the regulation), but another criteria like future likelihood 52 of re-election. Regardless of cause, reflecting the preference of coalition is critical for more accurate 53 evaluation of each player's importance in a game. 54

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Based on this notion, we introduce Preference Banzhaf, which computes Banzhaf value given each feature's probability of forming a coalition. We show that the attribution values can be computed through a centered (and later a regular) linear regression with binary masks, prove the convergence rate of the value, and empirically demonstrate the benefits of preference reflection. Our contributions are as follows:

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We introduce Preference Banzhaf, a novel algorithm that efficiently computes axiomsatisfying attribution using a different coalition-forming probability for each feature

We show the equivalence between Preference Banzhaf and (a) a centered linear regression shifted by each feature's probability, and (b) a regular linear regression with intercept

We derive the theoretical convergence rate of Preference Banzhaf

We empirically demonstrate the effect of using Preference Banzhaf and interpret what the different weights mean intuitively

2 Related Work

2.1 Model-Agnostic Explanations

Model-agnostic explanations usually involve perturbing the input and measuring the change in the 69 output. A fundamental method in this category is LIME [7], which fits an interpretable model 70 with kernel-weighted loss. The original method uses a linear model with a radial basis function 71 (RBF) kernel, but other kernels (such as cosine similarity kernel in Captum [19]) can be used. 72

Studies building upon LIME usually upgrade the sampling scheme or kernel selection. [9] trains a 73 causal model for generating perturbations, and [20] uses a clustering model to select perturbations 74 deterministically from the training dataset. [8] reformulates LIME as a Bayesian model to adjust the 75

LIME coefficients by some prior. [21] adopts an empirical pipeline to measure optimal RBF kernel 76 width for a desired level of local goodness of fit. [10] shows equivalence between RBF kernel and 77 adjusted feature mask probability, significantly stabilizing the attribution results by removing the 78 kernel from the regression. 79

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2.2 Game-Theoretic XAI

Game theory-based XAI literature focuses on developing methods that satisfy certain axiomatic properties. They tend to use Shapley value [11] (Equation 1) as the basis, which satisfies four properties: linearity, dummy, symmetry, and efficiency. While the Shapley value is too costly to calculate exactly, [12] shows that it can be estimated using a linear regression, a method known as KernelSHAP. The method has been adapted in many different directions [16], such as architecture specialization [22, 23, 24] or estimation method improvements [25, 26]. One issue with Shapley value is that it can be numerically unstable and difficult to compute in practice. Recent works relax some of the axioms - mainly efficiency - to address these shortcomings. For example, [27] propose Beta Shapley, which adjust the Shapley averaging scheme to include a Beta distribution.

A growingly popular alternative is Banzhaf value (Equation 2). While similar in construction to Shapley value, they differ in the treatment of the order of feature subsets. For Shapley value, the order is important: a set of size s that includes i as the m -th element is different from that as the l -th element, assigning different weights to the two coalitions. Banzhaf value considers both sets to be the same and simply averages across all possible subsets. Despite this difference, the two values are extremely similar, especially in terms of the rank of contributions [28, 29].

Most papers that use Banzhaf value often use regular Banzhaf value. [29] uses Banzhaf value for data valuation; [30] uses Shapley and Banzhaf value to select the optimal vocabulary subset for NLP tasks; and [31] utilizes Banzhaf value to create counterfactuals in graph neural networks. [32] generalizes Banzhaf value to weighted Banzhaf value for data valuation and shows that optimal weight w is dependent on the dataset and model. However, there has not been any research on computing Banzhaf values when all features have different weights, especially without relying on feature-wise calculations (referred to as Maximum Sample Reuse).

3 Method

3.1 Definition

Given players i ∈ S ⊆ N , let v ( S ) be the target value function for subset S . Let w i be the probability that player i joins a coalition, i.e., their coalition preference . Then, the Preference Banzhaf value ψ i p of player i is defined as:

Intuitively, ψ i p is the expected change in v given that each player may join the coalition following a multivariate binomial distribution with parameter w = { w 1 , w 2 , ..., w d } . Regular Banzhaf value is a special case where w i = 0 . 5 ∀ i , while weighted Banzhaf value is another special case where w i = α ∀ i .

3.2 Preference Banzhaf Approximation with Centered Linear Regression

KernelBanzhaf [33] approximates the Banzhaf value by masking each feature with probability w = 0 . 5 , and regressing the results against z = {-0 . 5 , 0 . 5 } d , where z i = -0 . 5 if x i is masked and 0.5 otherwise. This formulation can be generalized to any set of w i by using centered linear regression:

Theorem 1. Preference Banzhaf as Centered Linear Regression. Preference Banzhaf ψ p is the solution of the centered linear regression:

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一

where z i = m i -w i , p ( m i = 1) = w i , m i = 0 , 1 .

We can further show that the solution is still ψ p after adding an intercept term.

Theorem 2. Preference Banzhaf as Centered Linear Regression with Intercept. Preference Banzhaf ψ p is the solution of the centered linear regression with intercept:

The full proof for Theorems 1 and 2 are presented in the Appendix.

A consequence of Theorem 2 is that in terms of implementation, we do not need to center z to approximate the Preference Banzhaf value since centering does not affect the coefficients of a linear model when an intercept exists. We may perform the linear regression directly.

3.3 Convergence to True Value

A key question associated with kernel approximation of Banzhaf values is the rate of convergence to the true value. In the case of Preference Banzhaf value, it is closely related to GLIME [10] in implementation. Consequently, we can provide similar convergence guarantees.

Theorem 3. Convergence of Preference Banzhaf Assume that Z ∼ { b i -w i } d , where b i ∼ Ber ( w i ) . Then, given an empirical sample Z n and corresponding values v n , the linear regression solution β n converges to ψ p with probability 1 -δ (i.e., P ( | β n -ψ p | 2 ≤ ϵ ) ≤ 1 -δ

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for n = Ω( ϵ -2 M 2 v 2 d 3 γ 4 log (4 /δ ))) for some constant M , where v 2 = max ( w i (1 -w i )) and γ 2 = ∑ d i =1 1 / ( w i (1 -w i )) .

The full proof for convergence is presented in the Appendix. This theorem implies that, with all else 136 held constant, the solution converges the fastest when w i (1 -w i ) is maximized at w i = 0 . 5 , i.e., the 137 regular Banzhaf value. It also implies that weighted Banzhaf values with w i = α and w i = 1 -α 138 should have equal convergence under identical conditions. 139

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3.4 Synthetic Experiment for Convergence

Figure 3.4 shows the plots L 2 error of Preference Banzhaf estimates against v 2 and γ 4 for synthetic 141 datasets. Each subplot contains estimates for a model with 4 to 7 input features. The first row shows 142 the relation between L 2 error and v 2 γ 4 for random w applied on random quadratic functions, while 143 the second row shows the effect of γ 4 when v 2 is held constant at 0.25 (i.e., at least 1 w i =0.5). We 144

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4 Experiment 149

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4.1 Setup

Algorithms. We use the following algorithms for the experiments: 151

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KernelBanzhaf ( Kbanzhaf ) [33]: this is equivalent to setting w i = 0 . 5 .

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Weighted Banzhaf with probability α ( WBanzhaf ( α ) ): this is equivalent to setting w i = α . We use α of 0.25 and 0.75 to test the effect of α on convergence and explanation quality.

KernelSHAP [12] ( KernelSHAP ): this method approximates Shapley value using linear regression with combinatorial kernel.

MRLN [34]: We use this method to choose w i for Preference Banzhaf. Model Response Localized Attribution (MRLN) computes the empirical probability by sorting the samples by a distance metric from the original instance and averaging the mask of the closest samples. We follow the original paper for the best empirical thresholds.

Models and datasets. We train an XGBoost classifier for several datasets (Adult Census, Communities and Crime, California Housing, Diabetes, and Bank Marketing). We use the default settings from training the classifiers. Each model is trained on an random 80% split of the corresponding dataset.

Settings. For the Adult and Diabetes datasets, which have only 8 features, we generate explanations with maximum sample size equal to 2 d . For the rest of the tabular datasets, we use 500, 1000, and 2000 samples to evaluate the explanations. For tabular datasets, the evaluation is performed across 40 different seeds between 0 and 800. The replacement value for masking is a random instance in the opposite class. For image datasets, we use a baseline of 0 with a fixed seed of 0. The images are segmented into 64 equal segments. All evaluations are performed on the remaining 20% test split.

Faithfulness. We evaluate the faithfulness of the attributions using Area over Perturbation Curve [35] with predicted class's logit ( AOPC L ) and probability ( AOPC P ), as well as Iterative Removal of Features (IROF) [36]. It should be noted that while there are discussions on biases with these metrics [37, 38, 39], they are still widely used in the XAI literature for evaluation and it is outside of the scope of the study to discuss their limitations.

Sensitivity. We evaluate the sensitivity of the attributions using L2-normalized error [33], average 175 pairwise rank correlation, and top-K Jaccard index. For The last metric, we set K to 5 for Adult and 176 Diabetes datasets, and the minimum between 20 and half of the number of features for the rest of the 177 datasets. The sensitivity is evaluated only for tabular data due to computational constraints. 178

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4.2 Quantitative Evaluation

4.2.1 Sensitivity

The L 2 -normalized error for real datasets over N is presented in Figure 4.2.1. It is immediately obvious that Kbanzhaf achieves the lowest sensitivity amongst Banzhaf values, which agrees with Theorem 3 and the synthetic results since it minimizes v 2 γ 4 . KernelSHAP achieves lower sensitivity than WBanzhaf (0 . 75) in most datasets, but often loses to the other methods. MRLN is surprisingly robust, achieving third or second lowest L 2 -normalized error across all datasets. As will be shown in the subsequent section, MRLN also achieves higher average faithfulness than other methods, which suggests that we may generate high fidelity explanations with small robustness tradeoff by adjusting w i to a model's internal behavior. The sensitivity measured using Jaccard distance and correlation index (reported in Appendix C) also agree with that using L 2 -normalized error.

4.2.2 Faithfulness

The faithfulness evaluation of experiments on real tabular datasets is reported in Table 1. We can observe several patterns:

Excluding Preference Banzhaf with MRLN setup, the average faithfulness is generally the highest for regular Banzhaf value.

The average standard error of faithfulness (the standard deviation of a metric for each instance divided by square root of number of seeds, averaged across instances) follows similar order as sensitivity: generally, KBanzhaf is the smallest, followed by WBanzhaf (0 . 25) or MRLN , then KernelSHAP and WBanzhaf (0 . 75) .

The average standard error of faithfulness for WBanzhaf (0 . 75) tends to be much larger than the others. In particular, the average standard error for MRLN is comparable to WBanzhaf (0 . 25) despite the additional randomness caused by probability estimation.

These patterns demonstrate the effectiveness of using properly adjusted probabilities for Banzhaf values: we can achieve high and stable average fidelity.

4.3 Qualitative Evaluation

In this section, we analyze samples from image datasets to investigate the information captured by w i . Specifically, we compare the faithfulness of explanations depending on the location of high w i with respect to the true object in the image. Given a 64 equally divided segmentation map, we use w i = 0 . 7 (high weight) and w i = 0 . 3 (low weight) and either place the higher weight in the center 4 × 4 segments ( Banzhaf ( Center ) ) or the remaining periphery segments ( Banzhaf ( Periph ) ). Comparing the faithfulness between the two setups, we find the following patterns:

In terms of average faithfulness, Banzhaf ( Center ) has much higher fidelity than Banzhaf ( Periph ) as shown in Table 2. Given that many images in the Imagenette and Imagewoof datasets have their objects at the center of the image, this result implies that a higher overlap between the object and w i results in more faithful attributions.

This pattern coincides with instance-level differences. In Figure 4.3, we have examples where Banzhaf ( Center ) has much higher fidelity metric than Banzhaf ( Periph ) and vice versa. We see that when faithfulness of Banzhaf ( Center ) is higher, the main object is usually at the center. In the opposite case, the object is off-center or is too small compared to the window size.

This trend suggests that, to generate more faithful explanations, we need to select w i that is effectively the 'attention' of the model: higher w i should be assigned to features that the model focuses on for its predictions. It also explains why MRLN has higher average fidelity than other methods: it dynamically selects w i that aligns with the 'attention' of the model based on the target model's internal behavior. Note that the interpretation of w i is slightly different from an attribution, which determines how much (in positive or negative direction) a segment contributes to a prediction. w i only implies that the segment is important - we do not know the direction of said importance.

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5 Conclusion

In this paper, we present Preference Banzhaf, where each input feature is masked following a different probability, i.e., their preference of forming a coalition. We prove that Preference Banzhaf values can be computed through (a) a centered linear regression without intercept, and (b) a regular linear regression with intercept. We also derive the theoretical convergence given a set of preferences. We compare the faithfulness and sensitivity of MLRN-based Preference Banzhaf against different model-agnostic baseline methods across several tabular and image datasets. We find that Preference Banzhaf achieves the best average fidelity across all datasets, often followed by vanilla Banzhaf values. In terms of sensitivity, vanilla Banzhaf achieves the lowest sensitivity across all datasets, but is usually closely followed by Preference Banzhaf.

6 Limitations and Future Directions

There are several limitations to this work. Firstly, this paper focuses on accurately computing 239 Preference Banzhaf values given w i . Discovering methods of finding optimal w i for a given objective 240 using the relation between Preference Banzhaf and linear regression would be interesting. Secondly, 241 Preference Banzhaf is limited to fixed w i . Finding a fuzzy equivalent could help extend game242 theoretic XAI to more diverse set of model-agnostic explanations. Lastly, this research focuses solely 243 on feature attribution task. Extending Preference Banzhaf to other tasks such as data valuation could 244 show the benefits of using more generalized forms of game-theoretic XAI in different applications. 245

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Muhammad Rehman Zafar and Naimul Mefraz Khan. Dlime: A deterministic local interpretable model-agnostic explanations approach for computer-aided diagnosis systems. arXiv preprint arXiv:1906.10263 , 2019.

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353

A Experimental Details

The XGBoost classifiers are trained with default parameters from the xgboost package, while the 354 image classifiers are fine-tuned from IMAGENET10K weight available in the torchvision package. 355 The classification layer of the image classifiers consist of 4 linear layers with 20% dropout, batch 356 normalization, and ReLU activation. All training and experiments are performed on Intel(R) Xeon(R) 357 Gold 6342 CPU @ 2.8GHz and NVidia RTX A6000 (48GB). 358

B Proofs 359

In this section, we present the full proofs for theorems 1 through 3. 360

B.1 Proof for Theorem 1 361

Expanding the objective, we have: 362

̸

x i are independent Bernouilli variables with probability w i , which means Cov ( x i , x j ) = 0 . 363 Therefore, if we center x i so that E ( x i ) = 0 , i.e., subtract w i , then E ( x i x j ) = Cov ( x i , x j ) + 364 E ( x i ) E ( x j ) = 0 . 365

Then, the equation changes to: 366

which is equivalent to minimizing β pref,i individually. Taking the derivative for a single β pref,i , we 367 have: 368

Since E [ x 2 i ] = V ar ( x i ) = w i (1 -w i ) and E [ x i f ] = w i (1 -w i ) E [ f | x i = 1 -w i ] + (1 -369 w i )( -w i ) E [ f | x i = -w i ] : 370

Since x i = 1 -w i means feature i is included in the input set S and x i = -w i means it is excluded 371 from S , the above equation becomes: 372

̸

̸

B.2 Proof for Theorem 2 373

Equation 6 is identical to 5 except that we have the intercept term β 0 . Expanding the equation, we 374 have: 375

̸

̸

Since centering sets E [ x i ] = 0 and E [ x i x j ] = 0 : 376

Since the objective is equivalent, the solution stays identical as that from Equation 5. 377

B.3 Proof for Theorem 3 378

This proof closely follows the convergence of GLIME [10]. Since Preference Banzhaf is the solution 379 for a linear regression model, we know that: 380

where X n is the centered sampled masks and y n is the corresponding model predictions. Representing 381 Σ n = X T n X n and Γ n = X n y n , we would like to find the convergence of Σ -1 n Γ n to the limit Σ -1 Γ . 382 First, we can find the limit for Σ n as: 383

E ( X T X ) is equal to the variance of X since X has been centered, i.e., E ( x i ) = 0 ∀ i , which makes 384 Cov ( x i , x j ) = E ( x i x j ) -E ( x i ) E ( x j ) = E ( x i x j ) . Note that 0 ≤ σ 2 i ≤ 0 . 25 since each mask 385 follows a Bernouilli distribution. We can also bound the values of Σ n as follows: 386

̸

Therefore, all elements of || Σ n -Σ || are bounded to [ -0 . 25 , 1] , and we may apply matrix Hoeffding's 387 inequality with v 2 = max ( σ 2 i ) : 388

|| Σ -1 || 2 F is simply the sum of inverse of variances ∑ d 1 /σ 2 i = γ 2 . Lastly, we may apply Hoeffding's 389 inequality to Γ n to find: 390

Following [10], if we let n be the maximum among n 1 = 32 γ 2 v 2 log (4 d/δ ) , n 2 = 391 32 ϵ -2 M 2 d 2 γ 2 log (4 d/δ ) , and n 3 = 32 ϵ -2 M 2 v 2 dγ 4 log (4 d/δ ) , we have P ( || Σ -1 n Γ n -Σ -1 Γ || ≤ 392 1 -δ ) . 393

C Sensitivity 394

The sensitivity results using Jaccard distance and correlation index are as follows. The results agree 395 with that in the main figure with L 2 -normalized error. 396

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper advocated for weighted Banzhaf values, where each player $i \\in N$ joins a coalition with probability $w_i$. The theoretical analysis of these values has been carried out in [A], where this paper reproduces some of the results in Theorem 1 and 2. The paper proposes \"Preference Banzhaf\" as a family of weighted Banzhaf values, which are determined by finding the \"preferences of joining a coalition\" $w_i$. The empirical section evaluates these weighted Banzhaf values with $w_i \\equiv 0.75, w_i \\equiv 0.25$, and $w_i \\equiv 0.5$, as well as weights $w_i$ found by a \"MRLN\", a method not further described nor referenced. The empirical analysis compares these weighted Banzhaf values with approximated Shapley values from KernelSHAP on an XGBoost model and on ResNet101 using faithfulness metrics, such as perturbation curves and iterativel removal of features. The methods show comparable performance with the Shapley value  and KernelSHAP on several benchmark datasets.  MRLN achieves mostly superior results. Moreover, \"sensitivity\" is measured as well as qualitative examples in a CV setting.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper seems to be in preliminary stage with substantial flaws. The theoretical results have limited novelty, the experimental details are unclear in several aspects, such as implementation details and evaluation. While applying weighted Banzhaf values complementary to Shapley values could be an interesting direction of research, the benefits and design choices, e.g. finding the probabilities $w_i$, are not well explored.\n\n**Strengths**\n- The paper proposes an interesting concept to include the probability of a player joining the coalition in the computation of game-theoretic attribution scores.\n\n**Weaknesses**\n- The theoretical results, especially Theorem 1 and Theorem 2 are mostly known. Theorem 1 and the weighted least-squares representation of weighted Banzhaf values is well known, and has been proven, e.g. in [A]. Moreover, Theorem 2 follows from the dummy axiom, i.e. constant shifts in the value function do not affect the weighted Banzhaf values.\n- It remains unclear what the interpretation of the probabilities $w_i$ would be in practice. I was unable to find reference [34] online, which seems to be at the core of finding these probabilities with the MRLN method. Moreover, a key limitation of the method is that the probabilities are chosen independently per feature/player, which might be unreasonable.\n- The experimental section is quite chaotic. Additional insights into the computed metrics or additional plots, e.g. for AOPC, IROF metrics, would be helpful in the appendix. In the sensitivity plots, I would expect standard deviations and I was unable to follow what kind of sensitivity is measured here. The improvements seem overall quite minor, and implementation details are unclear, e.g. for MRLN. It is also unclear to me what should be observed from Figure 4.\n\n**Minor**\n-  Figures are generally wrongly referenced, e.g. \"Figure 4.2.1\" in line 181, \"Figure 3.4\" in line 141, \"Figure 4.3\" in line 216\n\n[A] Marichal, Jean-Luc, and Pierre Mathonet. \"Weighted Banzhaf power and interaction indexes through weighted approximations of games.\" European journal of operational research 211.2 (2011): 352-358.\n\nQUESTIONS:\n- Is it reasonable to choose independent probabilities per player? What is the interpretation of these probabilities? I was unable to follow how you identify these probabilities and what are their interpretation? Could you describe MRLN?\n- Could you provide more insights in to the computed metrics and their purpose, possibly in the appendix?\n\nLIMITATIONS:\nThe authors discuss limitations of their work.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces Preference Banzhaf, a novel game-theoretic feature attribution method that addresses limitations of existing approaches by relaxing the assumption that all feature coalitions are equally likely, and theoretically analyze the convergence properties of Preference Banzhaf. Extensive experiments on tabular and image datasets demonstrate the method's effectiveness.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The proposed Preference Banzhaf addresses the limitation of uniform coalition assumptions in traditional methods.\n- The paper establishes a theoretical equivalence between Preference Banzhaf and centered/regular linear regression, providing a fresh perspective on feature attribution.\n- Extensive experiments on tabular and image datasets demonstrate the method's effectiveness.\n\nWeaknesses:\n- The paper motivates Preference Banzhaf by drawing an analogy to real-world games where players have preferences for coalition formation. However, this analogy is not clearly justified in the context of machine learning. In ML, features are not autonomous agents with preferences; their \"coalition probabilities\" (i.e., w_i) lack a principled interpretation. The paper does not convincingly address why such feature-specific probabilities should exist or how they align with the goals of model explanation. This raises questions about whether the problem being solved is truly relevant to XAI.\n- A critical challenge in applying Preference Banzhaf is the difficulty of obtaining valid w_i values for real-world models. While the paper proposes using MRLN to estimate w_i, this approach relies on heuristics and does not provide theoretical guarantees for its validity. The experiments show improved performance when w_i is manually set to focus on central image regions (Banzhaf(Center)), but this scenario is contrived and does not generalize to arbitrary models or datasets. The paper lacks a systematic discussion of how to reliably estimate w_i in practice, especially for high-dimensional or black-box models where feature interactions are complex.\n- In experiments, the compared baselines do not include recent advances in XAI (e.g., SHAP variants like Beta Shapley [1]). A more comprehensive comparison will strengthen the claims.\n\n[1] Kwon Y, Zou J. Beta shapley: a unified and noise-reduced data valuation framework for machine learning [J]. arXiv preprint arXiv:2110.14049, 2021.\n\nQUESTIONS:\n- Could you explicitly justify why feature-specific coalition probabilities are meaningful in ML explanations? A direct connection to model properties (e.g., feature redundancy, model architecture, or data distribution) would strengthen the case for Preference Banzhaf.\n- Could you discuss limitations and potential pitfalls of using MRLN or similar heuristics for w_i estimation? Maybe you can include ablation studies on the impact of different w_i settings (e.g., random vs. learned) that would help validate the robustness of the method.\n- Since you have mentioned recent advances of SHAP variants like Beta Shapley, could you include it as a baseline to compare to?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn explainable AI, Shapley Value and Banzhaf value arising from game theory are frequently used to quantify the importance of a feature (comparable to a player in game theory). The importance of a feature f is measured by taking all subsets (called coalitions) of other features and measuring the average marginal improvement in performance of models by adding f to those subsets. \n\nThis paper is based on the premise that not all subsets are equally likely, as \".. in real games, each player can have different preference for joining a coalition.\"\n\nWith this idea, the paper introduces Preference Banzhaf value, where each feature $i$ is associated with a probability weight $w_i$ of its affinity of joining a coalition. The definition (eq 4) is the natural definition with such a consideration. \n\nThe theoretical results showing the value as a solution to a regression problem and convergence closely follow those for KernelBanzhaf. I have not checked the proofs in detail, but do not see a reason to doubt them. \n\nExperimental results compare with existing Banzhaf and Shapley value based methods. The Weights $w_i$ are set using MRLN [34].\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n\nThe paper has good readability. It provides theoretical as well as experimental results. It is on a timely topic. \n\nWeaknesses: \n\nThe main weakness is that the point of the paper is unclear. What does it means for different features to have different affinity to join coalitions and why it should this improve explanations are not discussed. \n\nUnlike human players, features do not have personal preferences. Their utility is determined by the training algorithm. And the purpose of Banzhaf value is to elicit this utility. Manipulating the coalitions themselves (by assigning weights to features befroehand) seems circular and interfering with the objective. \n\nIn experiments, the weights $w_i$ seems to be computed using some technique in reference [34]. but I cannot find this reference, and it is not explained what this method does. \n\nReference [34] simply says: \nAnonymous. MRLN: Adjusting masking probabilities based on model response, 2025.\n\nThe experiment section is generally poorly written. The quantities being measured and their importance are not explained. Overall, I am unconvinced that the experiments are showing something useful.\n\nQUESTIONS:\n1. Can you explain what $w_i$ represents? Why is the affinity for forming a coalition a factor, and how can we know it? An example can help. \n2. What is this reference [34] that you are using?\n3. What is your measure of $w_i$ in experiments? \n4. What do you mean by Faithfulness? \n5. What is the qualitative evaluation section and Fig 4 showing?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces the so-called Preference Banzhaf, a game-theoretic/perturbation based XAI alternative (following the SHAP value line), in particular a modification of the solution concept Banzhaf (in cooperative game theory, dropping the efficiency axiom) w.r.t. preference/likelihood of the feature taking place in the coalition of features. Authors show that their approach shows  favourable performance compared to KernelSHAP (a common variation of SHAP) and other Banzhaf-based methods. These empirical results (synthetic + real world datasets) confirm the theoretical results that authors introduce, basically convergence to centred linear regression (w/ and wo/ the intercept), and the true values. Here, implicitly the work is also inspired by LIME in terms of linear regression.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- The direction of research is relevant, in particular, considering the influence of game-theory inspired XAI techniques, both in academia and the industry. So it would be significant direction.\n\nWeakness: \n\nThere are plenty weaknesses:\n\nW1: In terms of clarity, lots of details are not nicely explained, especially formalisms and crucial technical values. See detailed comments below.\nW2: I wonder how theorem 1 and theorem 2 are not redundant, since one is with intercept (why do you need another theorem for this. )\n\nW3: regarding originality and quality, contribution seems to be incremental for this type of venue. \n\n\n\nMore detailed comments are as follows by my comments enumerated through Cs:\n\nC1: the motivation of their direction 46-54 is framed in terms of game theory solely, but it should be based on XAI, since you are not contributing to game theory but XAI. See Q1 below. \n\nC2:  in line 44 you don't need to say excluding said player, since this is the same for even classical Shapley value. (since it measures the player contribution). But you need to explain the Banzhaf value better. \n\nC3: In related work, you have a section on LIME, but how is this relevant  for your work (my guess, centred linear regression?) but this is not motivated. Also KernelBanzhaf seems to be highly relevant related work for your work, but you don't cite it in related work section, but only in methods. \n\nC4: Lots of technical details are not explained greatly.  w values,  GLIME,  what does these v^2 \\gamma^4 mean, and almost all the other constants/variables. T/here is basically no intuition provided. (e.g., M is just some constant, is it negative? is it positive? why it exist? what is a good value? is it important? how influential it is etc.) Similar for d delta choice of epsilon w_i, the powers of these terms etc.  \n\nC5: I am having hard time to justify why is this an important contribution, and you don't seem to explain it much either. Given the results, there is not much added value, then conceptually what should be the justification is not clear to me.  (Side comment: In the tables, you did not show Preference Banzhaf, but MLRN, which is also a bit confusing) These tables are not very informative.  Also, I don't know what to do with Figure 4, do they look great? (also mentioned in Question 3). In its current form, these look pretty weak in being informative (as the major rest of the paper).\n\nMinor issues:\n\n \n\nm1: typo in 39: \"KernelSHAP is suffers...\"\n\nm2: \"One issue with Shapley\" (I guess you mean Shapley value for KernelSHAP, must be indicated)\n\nm3: typing up the paranthesis of Equation 4 (preferably not squared ones but bigger normal parentheses)\n\n\nOverall, my warm recommendation to authors would be, to submit their work to dedicated decent venues (such XAI world conference), where the value of their contribution could be better appreciated, and hopefully get also further constructive feedback.\n\nQUESTIONS:\nQ1: What would these preferences correspond in XAI setting? (since features themselves cannot \"want\" to be with some other features). Are redundant or proxy features (always some taking place together) occurring with each other be considered as an example? But then how  can we justify the so-called \"preferences\"? (As in some case, I would argue that different coalitions would make a better coverage for the feature importance, so stop say, should we consider rather low probability ones?).\n\nQ2: Related to earlier question: how do we get these preferences in XAI setting right? You are using MLRN, but what does convergence to True value  mean?\n\nQ3: How do you respond to my comment 5 (C5). Also, I don't know what to do with Figure 4, do they look great?  (columns labels etc also missing, only explained in the caption, what is the legend? etc.  )\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 1, "ORIGINALITY": 1, "DATE": "2025-07-06"}