Efficiency — The sum of Shapley values adds up to the total payoff for the full group (in our case, $280).
Symmetry — If two players interact identically with the rest of the group, their Shapley values are equal.
Linearity — If the group runs a lemonade stand on two different days (with different team dynamics on each day), a player’s Shapley value is the sum of their payouts from each day.
Null player — If a player contributes nothing on their own and never affects group dynamics, their Shapley value is 0.
I don’t think this is true. Consider an alternative distribution in which each player receives their full “solo profits”, and receives a share of each synergy bonus equal to their solo profits divided by the sum of all solo profits of all players involved in the synergy bonus. In the above example, you receive 100% of your solo profits, 30/(30+10)=3/4 of the You-Liam synergy, 30/(30+20)=3/5 of the You-Emma synergy, and (30/30+20+10)=1/2 of the everyone synergy, for a total payout of $159. This is justified on the intuition that your higher solo profits suggest you are doing “more work” and deserve a larger share.
This distribution does have the unusual property that if a player’s solo profits are 0, they can never receive any payouts even if they do produce synergy bonuses. This seems like a serious flaw, since it gives “synergy-only” players no incentive to participate, but unless I’ve missed something it does meet all the above criteria.
Ah, I was going off the given description of linearity which makes it pretty trivial to say “You can sum two days of payouts and call that the new value”, looking up the proper specification I see it’s actually about combining two separate games into one game and keeping the payouts the same. This distribution indeed lacks that property.
Your calculations look right for Shapley Values. I was calculating based on Ninety-Three’s proposal (see here). So it’s good that in your calculations the sum of parts equals the combined, that’s what we’d expect for Shapley Values.
Doh! Thanks for the clarification. I see I misunderstood you targeting Ninety-Three’s proposal about locking in a “more work” ratio.
For me, locking in the ratio of solo profits intuitively feels unfair, and would not be a deal I’d agree to. Translating feeling to words, my personally-intuitive Alice (A) and Bethany (B) story would go:
Alice is a trained watchmaker, Bethany makes robots. They both go into the business of watch-making.
Alone, Alice pulls in $10,000/day. Expensive watches, but very slow to make.
Alone, Bethany pulls a meer $150/day. Cheapo ones, but she can produce tons!
Together, with Alice’s expertise + Bethany’s robot automation, they make $150,000/day!
Alone, neither is able to compensate for their weakness: Alice’s is production speed, Bethany’s being quality. The magic from their synergy comes from their individual weaknesses being overwritten by the other’s strengths. Value added is a completely separate entity versus the ratio of their solo efforts; it simply does not exist unless they partner up. Hence, I must treat it separately and that difference in total value vs. the sum of their individual efforts rightfully should be divided equally.
The value/cost of doing business w/ others, perhaps?
I don’t think this is true. Consider an alternative distribution in which each player receives their full “solo profits”, and receives a share of each synergy bonus equal to their solo profits divided by the sum of all solo profits of all players involved in the synergy bonus. In the above example, you receive 100% of your solo profits, 30/(30+10)=3/4 of the You-Liam synergy, 30/(30+20)=3/5 of the You-Emma synergy, and (30/30+20+10)=1/2 of the everyone synergy, for a total payout of $159. This is justified on the intuition that your higher solo profits suggest you are doing “more work” and deserve a larger share.
This distribution does have the unusual property that if a player’s solo profits are 0, they can never receive any payouts even if they do produce synergy bonuses. This seems like a serious flaw, since it gives “synergy-only” players no incentive to participate, but unless I’ve missed something it does meet all the above criteria.
I don’t think this proposal satisfies Linearity (sorry, didn’t see kave’s reply before posting). Consider two days, two players.
Day 1:
A ⇒ $200
B ⇒ $0
A + B ⇒ $400
Result: $400 to A, $0 to B.
Day 2:
A ⇒ $100
B ⇒ $100
A + B ⇒ $200
Result: $100 to A, $100 to B.
Combined:
A ⇒ $300
B ⇒ $100
A + B ⇒ $600
So: Synergy(A+B) ⇒ $200
Result: $450 to A, $150 to B. Whereas if you add the results for day 1 and day 2, you get $500 to A, $100 to B.
Ah, I was going off the given description of linearity which makes it pretty trivial to say “You can sum two days of payouts and call that the new value”, looking up the proper specification I see it’s actually about combining two separate games into one game and keeping the payouts the same. This distribution indeed lacks that property.
I’m just learning this, please forgive me if I’m misunderstanding. I’m calculating your example differently though:
Day 1: (200 + (400-200-0)/2) = 300 to A (0 + (400-200-0)/2) = 100 to B
Day 2: (100 + (200-100-100)/2) = 100 to A (100 + (200-100-100)/2) = 100 to B
Day 1+2: (300 + (600-300-100)/2) = 400 to A (100 + (600-300-100)/2) = 200 to A
300+100 does equal 400, 100+100 does equal 200
Sum of parts does equal the combined?
Your calculations look right for Shapley Values. I was calculating based on Ninety-Three’s proposal (see here). So it’s good that in your calculations the sum of parts equals the combined, that’s what we’d expect for Shapley Values.
Doh! Thanks for the clarification. I see I misunderstood you targeting Ninety-Three’s proposal about locking in a “more work” ratio.
For me, locking in the ratio of solo profits intuitively feels unfair, and would not be a deal I’d agree to. Translating feeling to words, my personally-intuitive Alice (A) and Bethany (B) story would go:
Alice is a trained watchmaker, Bethany makes robots. They both go into the business of watch-making.
Alone, Alice pulls in $10,000/day. Expensive watches, but very slow to make.
Alone, Bethany pulls a meer $150/day. Cheapo ones, but she can produce tons!
Together, with Alice’s expertise + Bethany’s robot automation, they make $150,000/day!
Alone, neither is able to compensate for their weakness: Alice’s is production speed, Bethany’s being quality. The magic from their synergy comes from their individual weaknesses being overwritten by the other’s strengths. Value added is a completely separate entity versus the ratio of their solo efforts; it simply does not exist unless they partner up. Hence, I must treat it separately and that difference in total value vs. the sum of their individual efforts rightfully should be divided equally.
The value/cost of doing business w/ others, perhaps?
This seems unlikely to satisfy linearity, as A/B + C/D is not equal to (A+C)/(B+D)