Explainable AI with Shapley Values (Part 1: Game Theory)

Key Takeaways

Shelby value is a famous game theory concept to understand how shape the values are used in explainable AI let's first understand how it was used in Game Theory it was introduced in 1951 by Lloyd shapley in a Cooperative game with many players shapley value calculates how important is each player to the overall Corporation and what payoff can each player expect so that sounds a little vague what exactly are shapley values and how is it calculated you might have seen those equations showing up everywhere on Wikipedia and in books and some people might get confused by those equations no worries in this video I'm going to help you understand those equations and understand sharply values in Game Theory let's say we are playing a Cooperative game and there are n players we're trying to figure out each player's contribution to the game each player makes a certain contribution to the game let's say Player 1 contributes one point player 2 contributes two points and so on in mathematical terms let's call this contribution function new new one equals 1 New 2 equals 2. you might think well you got an answer already new is each player's contribution it's not that simple players can join forces and when they join forces they might contribute differently for example player 2 and player 3 might be interested in working together and forming a coalition let's call this Coalition s s equals player 2 player three player two and three together contribute 8 points mathematically we write it as new s equals eight if we add player one to this Coalition that is s Union player one and three of them contribute 10 points so how much Point does player one contribute here that'll be two points now we understand how much player one contributes to this Coalition how do we calculate player-wise contribution to the entire game we need to figure out all the possible coalitions and simply take a weighted average of playerwise contribution to all possible coalitions that sounds pretty straightforward in this example with four players in our example without player one there will be three players they can form eight different coalitions we have one Coalition size of zero which is empty meaning that there is no teammate there are three coalitions with size of one player two player 3 and player 4. three coalitions size up two player two three two four and three four and one Coalition size of three how do we know there are three Coalition size of one three coalitions size of two and one Collision sets of three it is because a three choose one is three three choose two is three and three choose three is one so a minus one choose the Collision size will give us the number of different combinations of each Collision size let's rewrite everything in curly bracket so now we're going to calculate player one's contribution to each of those collisions remember the way we calculate player one's contribution in each Coalition is the difference between the contribution of Coalition plus player one and the contribution of the Coalition let's just make up some numbers here then we can calculate a weighted average of those numbers where each denominator in the parenthesis is the number of combinations of each clinician size and the denominator 4 is the number of different sizes of collisions which is also the number of players So based on this equation we get the number 2.5 so that is the contribution of Player 1 to the game which is also the sharply value of player one with the same logic we can calculate the sharply values for each player that is how shareply values are calculated in Game Theory thank you