- This step isn't obvious and you'll have to look at the example to see where the 1/3 come from. The last step is to normalize by the number of predictors, so you divide by 4. Therefore, the Shapley Value for predictor A would be 1.4667 / 4 = 0.37. Then you'd repeat this process and find the Shapley Values for predictors B, C, and D
- A famous example of the Shapley value in practice is the airport problem. In the problem, an airport needs to be built in order to accommodate a range of aircraft which require different lengths of..
- The Shapley value is a solution concept in cooperative game theory.It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Prize in Economics for it in 2012. To each cooperative game it assigns a unique distribution (among the players) of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable.
- I introduce cooperative games and illustrate an example of calculating the Shapley value.This video corresponds to this Chapter of my Game Theory class: http..
- The interpretation of the Shapley value for feature value j is: The value of the j-th feature contributed \(\phi_j\) to the prediction of this particular instance compared to the average prediction for the dataset. The Shapley value works for both classification (if we are dealing with probabilities) and regression
- Example 1: Alice and Bob and both necessary to produce something which has value 1500. Alice is player 1, Bob is player 2. Value of coalition The Shapley value of player 1 is: 750. The Shapley value of player 2 is: 750 List of examples Random example. Other things I have done.

It's Matt again and what we're going to do now is, is actually sort of wrap up our discussion of, of coalitional games and allocating value on a set of individuals and we're going to do so by looking at a particular example which we can do some comparison, say, of the Core and the Shapley Value and, and see exactly what's going on. so let's, let's look at a very interesting one. so think about. S hapley values that un cover n on-linear dependenc ies (Sunnies) are, to the best of our knowledge, the only Shapley-based feature importance methods that falls into the model-independent category. In this category, feature importance scores attempt to determine what is a priori. important, in the sense of understanding the partial dependence structures within the joint distribution. Deep learning example with DeepExplainer (TensorFlow/Keras models) Deep SHAP is a high-speed approximation algorithm for SHAP values in deep learning models that builds on a connection with DeepLIFT described in the SHAP NIPS paper. The implementation here differs from the original DeepLIFT by using a distribution of background samples instead of a single reference value, and using Shapley. 12 = 12.5. This is the Shapley value: x 1 = 10.5 and x 2 = 12.5. EXAMPLE 2. Suppose that there are three players now and v({1}) = 100, v({2}) =125, v({3}) = 50, v({1,2}) = 270, v({1,3}) = 375, v({2,3}) = 350 and v({1,2,3}) = 500. Then we have the following table ** Recap Analyzing Coalitional Games The Shapley Value The Core**. Shapley Value. Theorem. Given a coalitional game (N,v), there is a unique payoﬀ division x(v) = φ(N,v) that divides the full payoﬀ of the grand coalition and that satisﬁes the Symmetry, Dummy player and Additivity axioms

** One example is that in the tree-based models which might give two equally important features different scores based on what level of splitting was done using the features**. The features which split the model first might be given higher importance. This is the motivation for using the latest feature attribution method, Shapley Additive Explanations So, what is the Shapley value for Ram, Abhiraj, and Pranav each? It is just the average of the marginal payout for each! For example, for Ram it is (800 + 240 + 180 + 150 + 180 + 800)/6 = 392. Similarly, for Abhiraj it is 207, and for Pranav, it turns out to be 303

- This video shows how to allocate a common cost to multiple users with the Shapley Value Cost Allocation Method. The Shapley Value Method requires you to fir..
- This video from Game Theory Online (http://www.game-theory-class.org) derives the Core and the Shapley Value for a particular example game, and thereby illus..
- However, the absolute
**values**shall be taken into consideration and not the original positive and negative**values**because, as explained by Scott Lundberg, Positive and negative**values**are important, but if you allow them to cancel each other out when measuring global feature importance then you will not be measuring the importance of the features for the model, but rather how different the. - Ending Note: Shapley Value in the Mathematical Form. Twenty-something years ago when I was a graduate student learning cooperative games and the Shapley value, I wasn't quite sure any real-world applications but I was drawn to the elegance of the Shapley value concept
- Before introducing SHAP, let take a look on Shapley value which is a solution concept in cooperative game theory. Let take a development team as a n example. Our target is going to deliver a deep learning model which needs to finish 100 line of codes while we have 3 data scientists (L, M, N). 3 of them must work together in order to deliver the project
- In this video I describe an important idea of cooperative game theory: The Shapley value. At 3:51 there's a mistake, the marginal contribution made by C is 3..

Super class. iml::InterpretationMethod-> Shapley. Public fields. x.interest. data.frame Single row with the instance to be explained.. y.hat.interest. numeric Predicted value for instance of interest.. y.hat.average. numeric(1) Average predicted value for data X. sample.size. numeric(1) The number of times coalitions/marginals are sampled from data X. The higher the more accurate the. The Shapley Value was developed by the economics Nobel Laureate Lloyd S. Shapley as an approach to fairly distributing the output /n return shapley_values. The sample data that we used in. The Shapley value of each player is the average of its marginal contributions across all differently sized subgroups. For example, the value of B is equal to 5 (see bottom row). It is calculated as the average of 4, which is its individual output, 6, which is the mean contribution it makes to subgroups of size two, and 5, which is its marginal contribution to the overall group

Shapley Value Calculator Example 9: A small Indian state with 10 million inhabitants spends $60 million to vaccinate 30% of their population. An NGO which would otherwise be doing something really ineffective, comes in, and by sending reminders, increases the vaccination rate to 35% Shapley Value regression is also known as Shapley regression, Shapley Value analysis, LMG, Kruskal analysis, and dominance analysis, and incremental R-squared analysis. Worked example The first step with Shapley Value regression is to compute linear regressions using all possible combinations of predictors, with the R-squared statistic being computed for each regression 64 The Shapley Value 1 Introduction The purpose of this chapter is to present an important solution concept for cooperative games, due to Lloyd S. Shapley (Shapley (1953)). In the ﬁrst part, we will be looking at the transferable utility (TU) case, for which we will state the main theorem and study several examples. After

This video from Game Theory Online (http://www.game-theory-class.org) defines the Shapley Value, a prominent way of dividing profits within a coalition based.. Shapley value is a model agnostic method, For example, for additonal models and for models based on trees. Third — let's get a model in R and Python. Let's write some code 8.00 - Shapley Value Functions Example 2 Step 2 - Teradata Vantage Teradata® Vantage Machine Learning Engine Analytic Function Reference prodname Teradata Vantage vrm_release 1.0 8.00 created_date May 2019 category Programming Reference featnum B700-4003-098K. Introduction

- The Shapley value not only has desirable properties, it is also the only payment rule satisfying some subset of these properties. For example, it is the only payment rule satisfying the four properties of Efficiency, Symmetry, Linearity and Null player. See : 147-156 for more characterizations. Aumann-Shapley value
- For example, the Shapley value of a weighted majority game is computed in O(n3) computations in Algaba et al [2003]. Another example is in Littlechild and Owen [1973], who show that the computational complexity is O(n) if the worth of a coalition equals the maximal worth of a single coalition player
- variance based Shapley values. The Shapley value provides an importance measure that avoids the two prob-lems mentioned above: It is available for any function in L2 of the appropriate domain and it never gives negative importance. Although Shapley value solves the conceptual problems, computational prob-lems remain a serious challenge (Castro.
- The local Shapley values sum to the model output, and global Shapley values sum to the overall model accuracy, so that they can be intuitively interpreted, independent of the specifics of the model. In what follows, we'll walk through an example data set and see how global and local Shapley values can be calculated, visualised, and interpreted

- The Shapley value Essays in honor of Lloyd S. Shapley Edited by Alvin E. Roth i The right of the University of Cambridge to print and sell a/I manner of books Henry VIII in 1534. for example, institutional, social, or historical features-not modeled by the characteristic function
- The Shapley value allows contrastive explanations. Instead of comparing a prediction to the average prediction of the entire dataset, you could compare it to a subset or even to a single data point. This contrastiveness is also something that local models like LIME do not have. The Shapley value is the only explanation method with a solid theory
- ed 33/3
- e.g. Randomly sample with replacement my experimental data 1000 times, compute the Shapley value with each sample, and then average those computations. And at that point, it's not hard to construct a confidence interval using the 5% and 95% percentiles of the computations
- Shapley values plot for observation from apartments set and random forest model. The green and red bars correspond to the contribution of the variable to the prediction. The green ones take positive values, i.e. increase the prediction values, while the red ones take negative values, i.e. decrease the prediction value

Cost sharing — Shapley Value Last time we looked at sharing value in such a way, that no subgroup of players would get more value if they would just be by themselves. Value (or cost) sharing with this property is said to be in the core. The main problem with the core as we remarked last time, is that it is empty in almost all interesting cases Shapley Value vs. Markov Model: what are differences between the best models of marketing attributions and how they actually work. Let's find out together Shapley value for the above game is (7/6,1/6,4/6). Now recall from the previous chapter that the core of the above game equals {(t,0,2 − t) | t ∈ [1,2]}. So this example shows that the Shapley value does not need to lie in the core. Also note that in this game the Shapley value treats player 2 better than any core allocation And then these 2 are each getting a weight of 1 6th each. Right?, so that gives us the total value of the Shapley Value, and that tells us what person 1 should be getting in this setting. you know lets take a look at our simple simpler example just with 2 individuals and try to figure out what exactley the Shapley Value gets 8 Shapley Additive Explanations (SHAP) for Average Attributions. In Chapter 6, we introduced break-down (BD) plots, a procedure for calculation of attribution of an explanatory variable for a model's prediction.We also indicated that, in the presence of interactions, the computed value of the attribution depends on the order of explanatory covariates that are used in calculations

This package contains a function which can calculate the Shapley value in a cooperative game. Also, it contains some examples to clarify how to use the function The Shapely Value was introduced in 1951 by Lloyd Shapley, whom the theory was named after. As a solution concept, the Shapely value is used in scenarios when the contributions of the actors that work cooperatively are unequal. The Shapely value aims to assign gains and costs to all actors fairly. A Little More on What is Shapley Value

* Example 1: Find the Shapley-Owen decomposition for the linear regression for the data in range A3:D8 of Figure 1*. Figure 1 - Shapley-Owen Decomposition - part 1. We first calculate the R 2 values of all subsets of {x 1, x 2, x 3} on y, using the Real Statistics RSquare function. These values are shown in range G4:G11 This QScript computes Shapley Importance Scores, normalized so that their absolute values add up to 100%.. Technical details. Shapley importance determines what proportion of R-square from a linear regression model can be attributed to each independent variable

when calculating Shapley values. When this is handled inadequately, it can leave us with counter-intuitive explanations for our models' behaviour. This work introduces causal Shapley values which provide a theoretical substan-tiated view on how causality should be taken into account when calculating the Shapley value 1. 게임이론 (Game Thoery) Shapley Value에 대해 알기위해서는 게임이론에 대해 먼저 이해해야한다. 게임이론이란 우리가 아는 게임을 말하는 것이 아닌 여러 주제가 서로 영향을 미치는 상황에서 서로가 어떤 의사결정이나 행동을 하는지에 대해 이론화한 것을 말한다. 즉, 아래 그림과 같은 상황을 말한다. For example, between values of 0.65 to 0.68 for , the scores for the second predictor variable X 2 are less than the Relative Weights of the third predictor variable X 3 but more than the Shapley scores for X 3 While Shapley values have been shown to be the most theoretically-grounded additive feature attribution explanation methods, the exact (or even approximate) computation of Shapley values Examples of a Shapley module with 2 inputs can be seen in Fig. 5 in the appendix For this data-point, the cp and oldpeak values had the most negative influence on the prediction. For more examples on how to use Lime check out the Tutorials and API section in the README file. Shapley Values. The Shapley value is a method for assigning payouts to players depending on their contribution to the total payout

The Shapley Value Regression: Shapley value regression significantly ameliorates the deleterious effects of collinearity on the estimated parameters of a regression equation.The concept of Shapley value was introduced in (cooperative collusive) game theory where agents form collusion and cooperate with each other to raise the value of a game in their favour and later divide it among themselves The Shapley Value approach provides more accurate measures of effect importance compared to an ANOVA approach, whose estimates are sensitive to the order in which effects are introduced; and also compared to HLM, whose estimates vary more across different samples drawn from the same population—increasingly so as sample size decreases In the theoretical literature, this problem is most commonly addressed by sampling methods - rather than going over all possible feature coalitions, estimate the Shapley value using a sub-sample of them (if you are interested in this see for example Castro et al. 2009, Castro et al. 2017 or Benati et al. 2019). Enter the SHAP python librar As you can see from my example dataset, the Shapley value method is relatively straightforward to implement, but it has a downside - Shapley values must be computed for every single marketing channel combination - 2^(number of marketing channels) in fact, which becomes unfeasible for more than about 15 channels Shapley value. The idea of SHAP to compute $\phi_i$ is from the Shapley value in game theory. To understand this idea, let us imagine a simple scenario of solving a puzzle with prizes. With Alice alone, she scores 60 and get £60. Bob comes to help and they scored 80. When Charlie joins, the three of them scores 90

Shapley values and LIME. The connection between the Shapley values and LIME is noted in Lundberg-Lee (2017), but the underlying connection goes back to 1988 (Charnes et. al.). To see the connection, we need to modify LIME a bit For example, T-bills having a Shapley value of -1.447 % imply that the portfolio standard deviation decreases by that amount as mean return increases. Comparing T-bills with the small company, shows that the former reduces risk, whereas the latter increases risk in the same proportion * The Shapley Values Loop End node collects these predictions and calculates an approximation of the Shapley Values for each feature target combination*. A note on collections and vectors These nodes support collection and vector columns such as List columns, Bit Vector and Byte Vector columns, in case of which each element of the position/vector can be treated as an individual feature For **example**, SHAP (SHapely Additive exPlanations) Otherwise, use a **Shapley**-**value**-based method. Any model trained using gradient descent is differentiable. For **example**: neural networks, logistic regression, support vector machines. You can use IG with these Shapley values. An alternative for explaining individual predictions is a method from coalitional game theory that produces whats called Shapley values (Lundberg & Lee, 2016). The idea behind Shapley values is to assess every combination of predictors to determine each predictors impact

Example. shapr supports computation of Shapley values with any predictive model which takes a set of numeric features and produces a numeric outcome. The following example shows how a simple xgboost model is trained using the Boston Housing Data, and how shapr explains the individual predictions The many Shapley values for model explanation. 08/22/2019 ∙ by Mukund Sundararajan, et al. ∙ 0 ∙ share . The Shapley value has become a popular method to attribute the prediction of a machine-learning model on an input to its base features. The Shapley value [1] is known to be the unique method that satisfies certain desirable properties, and this motivates its use Note: one nice property of SHAP contributions is that their sum + sample mean is the same as the prediction: > shap_values. sum + clf. tree_. value [0]. squeeze 22.905199364899673 > clf. predict (df [: 1]) array ([22.9052]) Below we'll figure out why that's the case. For now, let's start on computing those values by hand. Shapley values

- While Shapley values result from treating each feature independently of the other features, it is often useful to enforce a structure on the model inputs. Enforcing such a structure produces a structure game (i.e. a game with rules about valid input feature coalitions), and when that structure is a nest set of feature grouping we get the Owen values as a recursive application of Shapley values.
- Shapley computes feature contributions for single predictions with the Shapley value, an approach from cooperative game theory. The features values of an instance cooperate to achieve the prediction. The Shapley value fairly distributes the difference of the instance's prediction and the datasets average prediction among the features.</p>
- the Weighted-Aumann Shapley value coincides with the Aumann-Shapley value. In contrast to the Aumann-Shapley value, however, the Weighted-Aumann Shapley value is well-deﬁned even when the risk capital allocation function 2An example would be an insurance company that holds both annuities and death beneﬁt insurance. Both types of liabilities ar

* 10 The Shapley Value 73 Example 2*.3 Three player zero-sum game We have three players in this game. They can form coalitions. If a player cooperates with another player their total gain is 1 and the third player loses 1. If they don't cooperate, they all lose 1 Shapley sampling values are meant to explain any model by: (1) applying sampling approximations to Equation 4, and (2) approximating the effect of removing a variable from the model by integrating over samples from the training dataset tive game theory. The Shapley value de nes a unique payo scheme that satis es many desiderata for the notion of data value. How-ever, the Shapley value often requires exponen-tial time to compute. To meet this challenge, we propose a repertoire of e cient algorithms for approximating the Shapley value. We also demonstrate the value of each trainin

Above are simple examples about using and analyzing the Shapley values in R. To see how features can be fairly compared, read Gabriel Tseng's Interpreting complex models with SHAP values article on Medium. Try looking at Shapley values on your Random Forest outputs and see what you find! Thank you for reading The Shapley value can then be seen as the average of the values that each player receives if the players are entered in a random order. Let's walk through a simple example: suppose that we are running a campaign to promote a particular product or service and that we are only using two marketing channels This notebook provides a brief example comparing various implementations of Shapley values using Kaggle's Titanic: Machine Learning from Disaster competition. While the true focus of the competition is to use machine learning to create a model that predicts which passengers survived the Titanic shipwreck, we'll focus on explaining predictions from a simple logistic regression model * For example, in the code chunk below we take the sum of the absolute value of the Shapley values within each feature to construct a Shap-based feature variable importance plot: # Load required packages library (ggplot2) If you want Shapley values for new instances*. This example shows that the Shapley value may not be in the core, and may not be the nucleolus. Stéphane Airiau (ILLC) - Cooperative Games Lecture 7: The Shapley Value 5 There are jCj! permutations in which all members of C precede i. There are jNn(C[fig)j! permutations in which the remaining members succede i, i.e. (jNj-jCj-1)!. The Shapley.

Example 11.2 Compute the Shapley Values (Kernel SHAP Method) Kernel SHAP with linearExplainer. This section contains PROC CAS code. Note: Input data must be accessible in your CAS session, either as CAS tables or as transient-scope tables For example, in medical data, if you use systolic and diastolic blood pressure (both are correlated) to train a model, in such scenario permutation method not able to distinguish the feature importance. Machine Learning Model Explanation using Shapley Values. September 26, 2020 September 27, 2020

Assume that the ShapleyAddOnePlayer input and output tables have the names InputTable and OutputTable, respectively. To compute a table that contains the Shapley value of each player: Create a table that contains the weighted payoff produced by adding a player to each combination. For example: CREATE MULTISET TABLE str.. Shapley value. In Section 4 we give an axiomatic characterization of the family of weighted Shapley values - that is, we provide a list of properties of a solution which is satisfied by and only by weighted Shapley values. Shapley (1981) proposed also a family of weighted cost allocations schemes an * Relationship between the Core and the Shapley value Put simply*, none::: the Shapley value is normative the Core is something else (hybrid) when the Core is non-empty, the SV may lie inside or not when the Core is empty, the SV is still uniquely determined 35/5

For example, Kasaei et al. formulated a risk-constrained two-stage stochastic programming problem to minimize the VPP operating cost while maintaining the system power quality. In Shapley value method is a popular method of solving cooperative game based profit allocation problems I understand Shapley value in game theory is a means to capture the average marginal contribution of a player. (which the Wikipedia article mentions but only gives a formula for in the example), namely as the average of the marginal contributions over all orders in which the players could join the complete coalition We can do this thanks to the structure of tree-based models and properties of Shapley values, mainly additivity (meaning that SHAP value for the model being a forest is a sum of SHAP values for all its trees). To further ensure that our method works fast, R package treeshap integrates C++ implementation of the algorithm We can simply sample values from the missing dimension(s) independently to fill the N/As and average over them to estimate the expectation. Of course, this will only really work out if we actually have feature independence, otherwise we're likely to sample values from the missing dimensions that create unlikely data points that don't really represent the underlying distribution

Shapley values: Better than counterfactuals Example 1 & recap: Sometimes, the counterfactual impact exceeds the total value. Example 2: Sometimes, the sum of the counterfactuals is less than total value. Sometimes it's 0. Example 3: Order indifference Der **Shapley**-Wert (benannt nach Lloyd **Shapley**) ist ein punktwertiges Lösungs-Konzept aus der kooperativen Spieltheorie.Er gibt an, welche Auszahlung die Spieler in Abhängigkeit von einer Koalitionsfunktion erwarten können (positive Interpretation) oder erhalten sollten (normative Interpretation) Shapley Values for Explaining ML models Define a coalition game for each model input x to be explained Players are the features of the input Gain is the model prediction F(x) Feature attributions are the Shapley values of this game We call the coalition game setup for computing Shapley Values as the Explanation Gam

Shapley value as a measurer of shareholders decision power example, the core of the game corresponds to the set of outcomes at which the seller sells to the buyer with the higher reservation price, at some price between twenty and thirty euro, and no other transfers are made illustrate this on a real-world example. 2 Causal Shapley values In this section, we will introduce causal Shapley values and contrast them to other approaches. We assume that we are given a machine learning model f() that can generate predictions for any feature vector x The main drawback of using the Shapley regressions framework is the computational cost of calculating Shapley value decompositions. Depending on the application, this can be addressed via appropriate approximations or sampling procedures. Section 6 concludes. An inference recipe for machine learning models is summarised in Box 1 in the Appendix to Based on the axioms which characterize the Shapley value (Shapley, 1953) for cooperative TU games, there are apparently many ways to extend the Shapley value to games in partition function form (see, for example, Myerson (1977), Bolger (1989), Potter (2000)). Myerson (1977) derived an eﬃcient value whic Comparing the results: The two methods produce different but correlated results. Another way to summarize the differences is that if we sort and rank the Shapley values of each sample (from 1 to 6), the order would be different by about 0.75 ranks on average (e.g., in about 75% of the samples two adjacent features' order is switched)

Gale Shapley Algorithm is an efficient algorithm that is used to solve the Stable Matching problem. It takes O(N^2) time complexity where N is the number of people involved • the Shapley value when the threshold is Mequals the Shapley value when the weights are powers of 2, and the threshold is Ú # M • Computing the Shapley value for super‐increasing weights boils down to computing it for powers of 2! • Using this claim, we obtain a closed‐form formula o Shapley value can be derived from a set of axioms, although this can be done in more than one way (see, e.g., Aubin 1981; Billera and Heath 1982; Mirman and Tauman 1982). One aspect of the Aumann-Shapley value that appears to deviate qualitatively from the Shapley value Value. The shapleyValue functions returns a matrix with all the marginal contributions of the players (contributions) and a vector with the Shapley value (value). References. Lloyd S. Shapley. A Value for n-person Games. In Contributions to the Theory of Games, volume II, by H.W. Kuhn and A.W. Tucker, editors

SHAP (SHapley Additive exPlanation) leverages the idea of Shapley values for model feature influence scoring. The technical definition of a Shapley value is the average marginal contribution of a feature value over all possible coalitions. For example, SHAP has a tree explainer that runs fast on trees,. Data Shapley value uniquely satisfies several natural properties of equitable data valuation. We develop Monte Carlo and gradient-based methods to efficiently estimate data Shapley values in practical settings where complex learning algorithms, including neural networks, are trained on large datasets For example, the Shapley value can be used to determine what each member of a group should pay in a restaurant when everyone shares their food. The theory of Shapley values are based on four axiomatic principles, efficiency, symmetry, missigness, and additivity, which makes this method unique compared to other explanation techniques — even our beloved regression coefficients value [Shapley, 1953]. An intuitive example of the potential use of the Shapley value can be provided in an academic set-ting. Assume that youare a Professorrunninga lab, and, once and for all, you have decided to distribute the yearly bonus to your students in fair manner, that reﬂects the actual con Figure 6. Using the Shapley algorithm to measure the impact and direction of a feature. Red on the right of the SHAP value 0.00 means that the mortgage loan is more likely to become delinquent because of the feature value being higher. Blue on the right of the SHAP value 0.00 means less likely. Feature names typically appear on the left side

Shapley values for a linear-ensemble model can be computed as linear combinations of Shapley values for its constituent models. Axiom 3 (Nullity) guarantees that if a feature is completely disconnected from the model's output, it receives zero Shapley value. Axiom 4 (Symmetry) requires attribution t Request PDF | Shapley value | The value of an uncertain outcome (a 'lottery') is an a priori measure, in the participant's utility scale, of what he expects to obtain. ables whereas SPVIM would assign the same positive value to both variables. In this paper, we take advantage of an alternate formula-tion of the Shapley value noted in previous work (see, e.g., Charnes et al., 1988; Lundberg and Lee, 2017). In particular, we can rewrite the weighted average in (1) as the solution o When you add up Player C's marginal value from the 6 possible orders and divide it by 6, you get a Shapley value of 8.1. Now that we have worked out the Shapley value for each player, we can clearly see the true contribution each player made to the game and assign credit fairly. In this example, player C contributed the most, followed by A. Shapley Homology: Topological Analysis of Sam-ple Inﬂuence for Neural Networks Kaixuan Zhang1, Qinglong Wang2, Xue Liu2;C. Lee Giles1 1Information Sciences and Technology, Pennsylvania State University. 2School of Computer Science, McGill University. Keywords: Topological data analysis, homology, Shapley value, sample inﬂuence, deep learning

The Shapley value is commonly used in cooperative set-tings [14, 9, 19, 10, 17, 9] to evaluate participants. There are applications of the Shapley value in feature selection; e.g., [4] where they use a randomized sub-sampling method for approximating the Shapley value. Sometimes a domain has speci c structure that allows for the Shapley value to b I am currently working on a For loop in R. If I run the For loop on my own data, it takes ages, and I believe because I did something inefficient in my code. Could you please help me with improving.. 2.3. Shapley values Shapley values can be considered a particular example of perturbation-based methods where no hyper-parameters, ex-cept the baseline, are required. Consider a set of Nplayers Pand a function ^fthat maps each subset S Pof players to real numbers, modeling the outcome of a game when players in Sparticipate in it. Th