The calculation of OSWC is similar to the calculation of SRC but with two key differences.

First, the OSWC uses a different value function. The OSWC value function is calculated by simulating all half-innings in the game rather than just one half-inning. When the entire game is simulated for a coalition, only the event triggers for the teammates in that coalition are included in the simulation, and when the event triggers for one half-inning are completed, the base-out state resets to 000-0 before proceeding to the next event trigger in the sequence of plays.

Second, once the event triggers for that coalition have been traced through for all half-innings, if the coalition has produced more runs than the opponent scored in that game, then the value for that coalition is 1, but if the coalition did not outscore the opponent, then the value for that coalition is 0. In words, the coalition’s goal is to win by outscoring the opponent given the number of runs scored by the opponent.

By taking value 1 or 0 depending on whether the coalition outscored the opponent, the value function is understood to be in win units. Thus, a team that outscores the opponent has won the game, and the Shapley Value calculated using this win value function is appropriately understood as distributing the fair amount of credit for outsourcing the opponents among the teammates.

Let us return to Game 6 of the 2020 World Series. In the top of the 1st, Arozareno hit a solo HR for Tampa Bay. We already discussed the Dodgers scoring 2 runs in the bottom of the 6th inning. Betts also hit a solo HR in the bottom of the 8th inning to complete the scoring for the Dodgers 3-1 victory.

We follow similar steps here as we did for calculating SRC. We again must consider all of the possible coalitions for all of the Dodgers who played in the game. However, it turns out that all Dodgers except Barnes, Betts, and Seager are Null players, so let us remove them all but these three Dodgers to simplify the calculation example.

STEP 1: EVENT-TRIGGER SCOREKEEPING

As with the SRC calculation, we start with the event-trigger scorekeeping. In this case, we need all event triggers in the game for all of the teammates. With only Barnes, Betts, and Seager in the set of teammates, then we include only their plays.

Table 1 lists all of the event-trigger labels for Barnes, Betts, and Seager that were used on this website to calculate SRCC and OSWC. As before, we use A to represent [A]ustin Barnes, B to represent Mookie [B]etts, and C to represent [C]orey Seager. We will not discuss here why we chose these event triggers, but leave it up to you to examine the choices later on your own. Table 2 lists the event triggers.

**Table 1: Event-trigger Labels, Game 6, 2020 World Series**

Inning | Player | Event Trigger Label |

1 | M Betts | OUT |

1 | C Seager | OUT |

3 | A Barnes | GO_A |

3 | M Betts | OUT |

4 | C Seager | OUT |

6 | A Barnes | SGL_A |

6 | M Betts | DBL_A |

6 | A Barnes | SB_H |

6 | M Betts | SB_H2 |

6 | C Seager | FC_4 |

7 | A Barnes | FO_A |

8 | M Betts | HR |

8 | C Seager | BB |

**Table 2: Event Triggers, Game 6, 2020 World Series**

STEP 2: SIMULATE THE ENTIRE GAME FOR EVERY COALITION

We construct the value function by tracing through the event triggers for each of the possible coalitions using just Barnes, Betts, and Seager. Table 3 provides the value function that follows from this procedure.

**Table 3: Value Function, Dodgers with Full Coalition ABC, Game 6, 2020 World Series**

Coalition Members | Value Function |

None (the empty set) | 0 |

A | 0 |

B | 0 |

C | 0 |

AB | 1 |

AC | 0 |

BC | 1 |

ABC | 1 |

The None coalition has no event triggers, so no runs score. 0 runs is less than the 1 scored by the Rays, so the Dodgers’ win value for the None coalition in Table 3 is 0.

The coalition A has just Barnes’s event triggers. With only Barnes’s event triggers, the Dodgers score 0 runs. The 3rd inning has only Barnes’s groundout, so the 3rd inning ends with just a single groundout and 0 runs scored. The 6th inning has Barnes’s single and base advance on the wild pitch, so the 6th innings ends with a runner on 2nd with no outs but also 0 runs scored. The 7th inning has just a flyout, which yields 0 runs scored. The 0 runs that score in total is less than the Rays’ 1 run, so a coalition of just Barnes will lose the game. Thus, the win value for coalition A is 0.

The coalition B has just Betts’s event triggers. With just Betts’s event triggers, the Dodgers score 1 run. Inning 1 ends with an out but no runs. Inning 3 ends with an out and no runs. Inning 6 ends with Betts on third after a double and base advance on a wild pitch. Inning 8 ends with 1 run resulting from Betts’s HR. Because a Dodgers coalition of just Betts produces 1 run, the Dodgers do not have enough runs to win the game. So the win value for coalition B is 0.

The coalition C has just Seager’s event triggers. Inning 1 has only Seager making an out, so 0 runs scored. Inning 4 has just Seager making an out, so 0 runs scored. Inning 6 has Seager making on out because with FC_4 Seager is out at first when starting at initial state 000-0, so again 0 runs scored. Inning 8 has Seager getting a walk, but with no other events, the inning ends with 0 runs scored. With 0 runs scored, the win value for coalition C is 0.

The AB coalition with just Barnes and Betts can win the game, however. Inning 1 with just Betts produces 0 runs, and inning 3 has two outs made and 0 runs scored. But inning 6 with Barnes and Betts scores 1 run (as explained in How to Calculate SRC), and Betts’s HR in inning 8 produces 1 run. Thus, if the Dodgers have only Barnes’s and Betts’s event triggers, then they score 2 runs which is more than the Rays’ 1 run. Because the AB coalition outscores the Rays, it has win value 1.

By similar tracing, we are able to show that coalition BC has win value 1, coalition AC has win value 0, and the full coalition has win value 1. This completes Table 2.

STEP 3: CALCULATE THE SHAPLEY VALUES USING THE VALUE FUNCTION

Because the values in Table 3 are in win units, the Shapley Value calculation will distribute the credit for the 1 win that resulted from the full coalition of teammates. This fact allows us to interpret the Shapley Value for a player as that player’s OSWC, i.e., the credit to that player for their role in outsourcing the opponent when credit is fairly distributed among the players..

We now use the value function in Table 3 to calculate the Shapley Values for A, B, and C, which is done in Table 4, 5, and 6.

**Table 4: Shapley Value Calculation for A**

Order | Value before adding A: | Value after adding A | Marginal value of adding A |

A-B-C | 0 | 0 | 0 |

A-C-B | 0 | 0 | 0 |

B-A-C | 0 | 1 | 1 |

B-C-A | 1 | 1 | 0 |

C-A-B | 0 | 0 | 0 |

C-B-A | 1 | 1 | 0 |

**Table 5: Shapley Value Calculation for B**

Order | Value before adding A: | Value after adding A | Marginal value of adding A |

A-B-C | 0 | 1 | 1 |

A-C-B | 0 | 1 | 1 |

B-A-C | 0 | 0 | 0 |

B-C-A | 0 | 0 | 0 |

C-A-B | 0 | 1 | 1 |

C-B-A | 0 | 1 | 1 |

**Table 6: Shapley Value Calculation for C**

Order | Value before adding A: | Value after adding A | Marginal value of adding A |

A-B-C | 1 | 1 | 0 |

A-C-B | 0 | 0 | 0 |

B-A-C | 1 | 1 | 0 |

B-C-A | 0 | 1 | 1 |

C-A-B | 0 | 0 | 0 |

C-B-A | 0 | 0 | 0 |

Our Shapley Value calculation reveals that the fair distribution of credit for outsourcing the opponent is such that Barnes (A) has 0.167 dPW. Betts (B) has 0.667 OSWC, and Seager (C) has 0.167 OSWC.