The Core: A Different View on Explanation

This example introduces the game-theoretic concept of the core and the egalitarian least core (ELC), and shows how to compute them with ExactComputer.

from __future__ import annotations

import numpy as np

import shapiq

The Paper Game

Three AI researchers (Alice, Bob, Charlie) co-author a paper that wins a $500 best-paper award. We model their joint productivity as a cooperative game:

Coalition	Award
{}	$0
{Alice}	$0
{Bob}	$0
{Charlie}	$0
{Alice, Bob}	$500
{Alice, Charlie}	$400
{Bob, Charlie}	$350
{Alice, Bob, Charlie}	$500

Shapley values give: Alice=$200, Bob=$175, Charlie=$125. But Alice and Bob could earn $250 each by excluding Charlie – so the core is empty.

The Least Core and ELC

The least core finds the minimum subsidy \(e\) that makes cooperation stable. The egalitarian least core picks the distribution with minimum \(\|x\|_2\) from the least core.

class PaperGame(shapiq.Game):
    def __init__(self) -> None:
        super().__init__(n_players=3, normalize=True, normalization_value=0)

    def value_function(self, coalitions: np.ndarray) -> np.ndarray:
        values = {
            (): 0,
            (0,): 0,
            (1,): 0,
            (2,): 0,
            (0, 1): 500,
            (0, 2): 400,
            (1, 2): 350,
            (0, 1, 2): 500,
        }
        return np.array([values[tuple(np.where(x)[0])] for x in coalitions])


paper_game = PaperGame()

Computing the Egalitarian Least Core

exact_computer = shapiq.ExactComputer(n_players=3, game=paper_game)
elc = exact_computer("ELC")
print("ELC payoffs:", elc.dict_values)
print("Stability subsidy e*:", exact_computer._elc_stability_subsidy)

ELC payoffs: {(0,): 233.33333333333323, (1,): 183.33333333333337, (2,): 83.33333333333336}
Stability subsidy e*: 83.33333333333341

The ELC distributes approximately: Alice $233, Bob $183, Charlie $83, with a stability subsidy of about $83.