the shapley value in knaster's collusion game€¦ · the shapley value in knaster’s...

The Shapley Valuein Knaster’s Collusion Game

F. Briata1 A. Dall’Aglio2 M. Dall’Aglio3 V. Fragnelli4

1U. Genoa 2Sapienza U. Rome 3LUISS U. 4U. Eastern Piedmont

Game Theory and RationingLake Como School of Advanced Studies, September 6, 2018

M.Dall’Aglio (LUISS) Collusion and Knaster Campione D’Italia 2018 1 / 25

INTRODUCTION

Consider a group of agents/players who have claims over anindivisible object

Knaster (1946) devised a simple procedure to allocate the indivisibleitem among the players.

A fair allocation is obtained by selling the item to one of the playersand by distributing the resulting amount among all the players in afair way

The procedure is not immune to collusive behaviors

We analyze this collusive behavior when players make bindingagreements among themselves

INTRODUCTION

Consider a group of agents/players who have claims over anindivisible object

Knaster (1946) devised a simple procedure to allocate the indivisibleitem among the players.

A fair allocation is obtained by selling the item to one of the playersand by distributing the resulting amount among all the players in afair way

The procedure is not immune to collusive behaviors

We analyze this collusive behavior when players make bindingagreements among themselves

N = {1, 2, . . . , n} set of players;

bi valuations of item from player i ∈ N.

WLOG b1 ≥ b2 ≥ · · · ≥ bn

KNASTER’S PROCEDURE

(i) valuations are communicated to a mediator

(ii) Player 1 gets the object and pays b1;

(iii) Each player i ∈ N receives a monetary compensation Bi = Ei + Si ,where:

(a) Ei is the initial fair share

(b) Si is the equal share of the surplus

(b1 −

∑i∈N

N = {1, 2, . . . , n} set of players;

bi valuations of item from player i ∈ N.

WLOG b1 ≥ b2 ≥ · · · ≥ bn

KNASTER’S PROCEDURE

(i) valuations are communicated to a mediator

(ii) Player 1 gets the object and pays b1;

(iii) Each player i ∈ N receives a monetary compensation Bi = Ei + Si ,where:

(a) Ei is the initial fair share

(b) Si is the equal share of the surplus

(b1 −

∑i∈N

)M.Dall’Aglio (LUISS) Collusion and Knaster Campione D’Italia 2018 3 / 25

EXAMPLE

Player 1 2 3 4

bi 240 192 80 48

Ei 60 48 20 12 S = 100

Si 25 25 25 25

Bi 85 73 45 37

In practice: Pl.1 gets the item and pays 240− 85 = 155. In turn, theamount of 155 will be distributed among the other players who will receive73, 45 and 37 respectively.

PROPERTIES

The resulting allocation is proportional Bi ≥1

nbi i ∈ N

Knaster’s procedure does not require or produce money

The procedure can be easily extended to the allocation of severalitems: simply apply a separate procedure for each item

EXAMPLE

Player 1 2 3 4

bi 240 192 80 48

Ei 60 48 20 12 S = 100

Si 25 25 25 25

Bi 85 73 45 37

PROPERTIES

nbi i ∈ N

EXAMPLE

Player 1 2 3 4

bi 240 192 80 48

Ei 60 48 20 12 S = 100

Si 25 25 25 25

Bi 85 73 45 37

PROPERTIES

nbi i ∈ N

EXAMPLE

Player 1 2 3 4

bi 240 192 80 48

Ei 60 48 20 12 S = 100

Si 25 25 25 25

Bi 85 73 45 37

PROPERTIES

nbi i ∈ N

EXAMPLE

Player 1 2 3 4

bi 240 192 80 48

Ei 60 48 20 12 S = 100

Si 25 25 25 25

Bi 85 73 45 37

PROPERTIES

nbi i ∈ N

COLLUSIONIt is well known (see eg Brams and Taylor ’96) that the procedure ismanipulable: Players may gain by declaring a false valuation b′i 6= bi ,i ∈ NHowever, when player do not know the evaluation of others and theyare infinitely risk-averse, i.e. they manipulate only when they gainwith certainty, the procedure becomes non-manipulable

Players in a group S ⊂ N may decide to collude

PROPOSITION (Fragnelli and Marina ’09)

A subset of players S ⊂ N will obtain the highest safe gain when allplayers declare the highest true valuation in the set:

bS := maxi∈S

bi for any i ∈ S

A collusion of a subset of infinitely risk-averse players will consists of(i) Truthful declaration among them of their valuation;(ii) Binding agreement on the gain sharing;(iii) Same declaration of the highest true valuation.

COLLUSIONIt is well known (see eg Brams and Taylor ’96) that the procedure ismanipulable: Players may gain by declaring a false valuation b′i 6= bi ,i ∈ NHowever, when player do not know the evaluation of others and theyare infinitely risk-averse, i.e. they manipulate only when they gainwith certainty, the procedure becomes non-manipulablePlayers in a group S ⊂ N may decide to collude

bS := maxi∈S

bi for any i ∈ S

COLLUSIONIt is well known (see eg Brams and Taylor ’96) that the procedure ismanipulable: Players may gain by declaring a false valuation b′i 6= bi ,i ∈ NHowever, when player do not know the evaluation of others and theyare infinitely risk-averse, i.e. they manipulate only when they gainwith certainty, the procedure becomes non-manipulablePlayers in a group S ⊂ N may decide to collude

bS := maxi∈S

bi for any i ∈ S

GAIN AND COALITION STRUCTURE

EXAMPLE continued

1 2 3 4

true bi 240 192 80 48

Bi 85 73 45 37

{1, 2} coll 240 240 80 48

Bi 82 82 42 34

Gain of 1 and 2 when

noone else colludes =

2 ∗ 82− 85− 73 = 6

{3, 4} coll 240 192 80 80

Bi 83 71 43 43

{1, 2}, {3, 4} 240 240 80 80

Bi 80 80 40 40

3 and 4 collude =

2 ∗ 80− 83− 71 = 6

PROPOSITION

The gain of a coalition S is independent from the behavior of other players

EXAMPLE continued

1 2 3 4

true bi 240 192 80 48

Bi 85 73 45 37

{1, 2} coll 240 240 80 48

Bi 82 82 42 34

2 ∗ 82− 85− 73 = 6

{3, 4} coll 240 192 80 80

Bi 83 71 43 43

{1, 2}, {3, 4} 240 240 80 80

Bi 80 80 40 40

3 and 4 collude =

2 ∗ 80− 83− 71 = 6

PROPOSITION

EXAMPLE continued

1 2 3 4

true bi 240 192 80 48

Bi 85 73 45 37

{1, 2} coll 240 240 80 48

Bi 82 82 42 34

2 ∗ 82− 85− 73 = 6

{3, 4} coll 240 192 80 80

Bi 83 71 43 43

{1, 2}, {3, 4} 240 240 80 80

Bi 80 80 40 40

3 and 4 collude =

2 ∗ 80− 83− 71 = 6

PROPOSITION

EXAMPLE continued

1 2 3 4

true bi 240 192 80 48

Bi 85 73 45 37

{1, 2} coll 240 240 80 48

Bi 82 82 42 34

2 ∗ 82− 85− 73 = 6

{3, 4} coll 240 192 80 80

Bi 83 71 43 43

{1, 2}, {3, 4} 240 240 80 80

Bi 80 80 40 40

3 and 4 collude =

2 ∗ 80− 83− 71 = 6

PROPOSITION

EXAMPLE continued

1 2 3 4

true bi 240 192 80 48

Bi 85 73 45 37

{1, 2} coll 240 240 80 48

Bi 82 82 42 34

2 ∗ 82− 85− 73 = 6

{3, 4} coll 240 192 80 80

Bi 83 71 43 43

{1, 2}, {3, 4} 240 240 80 80

Bi 80 80 40 40

3 and 4 collude =

2 ∗ 80− 83− 71 = 6

PROPOSITION

THE GAIN GAME

DEFINITION

For any S ⊂ N the maximum gain from collusion of the players in S is

vg (S) =n − s

∑i∈S

(bS − bi ) ≥ 0

where |S | = s and bS := maxi∈S bi . vg is the gain game.

Each player outside the colluding coalition S suffers an identical loss

−vg (S)

n − s= −

∑i∈S

(bS − bi ) ≤ 0

vg ({i}) = 0 for every i ∈ N ⇒ No player colludes alone

vg (N) = 0 No external player to harm

THE GAIN GAME

DEFINITION

For any S ⊂ N the maximum gain from collusion of the players in S is

vg (S) =n − s

∑i∈S

(bS − bi ) ≥ 0

where |S | = s and bS := maxi∈S bi . vg is the gain game.

Each player outside the colluding coalition S suffers an identical loss

−vg (S)

n − s= −

∑i∈S

(bS − bi ) ≤ 0

vg ({i}) = 0 for every i ∈ N ⇒ No player colludes alone

vg (N) = 0 No external player to harm

WHY SHAPLEY?

We now measure the contribution of each player to the colluding process:

We consider the Shapley value of vg is the contribution of player i tothe coalitions averaged over all possible orders of entry

φi (vg ) =1

|N|!∑R

vg(PRi ∪ {i}

)− vg

)where the sum ranges over all |N|! orders R of the players and PR

i isthe set of players in N which precede i in the order R

Since∑

i∈N φi (vg ) = v(N) = 0, unless all values are zero, wedistinguish two groups

φi (vg ) > 0 =⇒ player i favours collusion (on average)φi (vg ) < 0 =⇒ player i discourages collusion (on average)

WHY SHAPLEY?

We now measure the contribution of each player to the colluding process:

We consider the Shapley value of vg is the contribution of player i tothe coalitions averaged over all possible orders of entry

φi (vg ) =1

|N|!∑R

vg(PRi ∪ {i}

)− vg

)where the sum ranges over all |N|! orders R of the players and PR

i isthe set of players in N which precede i in the order R

Since∑

i∈N φi (vg ) = v(N) = 0, unless all values are zero, wedistinguish two groups

φi (vg ) > 0 =⇒ player i favours collusion (on average)φi (vg ) < 0 =⇒ player i discourages collusion (on average)

Computing the Shapley ValueWe consider the difference of the Shapley value for two consecutive players

φj(vg )− φj+1(vg ) =∑S⊂N\{j ,j+1}

s!(n − s − 2)!

(n − 1)![vg (S ∪ {j})− vg (S ∪ {j + 1})] (1)

(s = |S |) together with ∑i∈N

φi (vg ) = vg (N) = 0

The application of (1) is made possible by the following

For any j ∈ N \ {n} and S ⊂ N \ {j , j + 1},

vg (S∪{j})−vg (S∪{j+1}) =

{(n−s−1)s

n2(bj − bj+1) if S ⊂ {j + 2, . . . , n}

−n−s−1n2

(bj − bj+1) otherwise

Computing the Shapley ValueWe consider the difference of the Shapley value for two consecutive players

φj(vg )− φj+1(vg ) =∑S⊂N\{j ,j+1}

s!(n − s − 2)!

(n − 1)![vg (S ∪ {j})− vg (S ∪ {j + 1})] (1)

(s = |S |) together with ∑i∈N

φi (vg ) = vg (N) = 0

The application of (1) is made possible by the following

For any j ∈ N \ {n} and S ⊂ N \ {j , j + 1},

vg (S∪{j})−vg (S∪{j+1}) =

{(n−s−1)s

n2(bj − bj+1) if S ⊂ {j + 2, . . . , n}

−n−s−1n2

(bj − bj+1) otherwise

EXAMPLE continued

The Shapley value for our example is

φ(vg ) =

Extreme players are more prone to collusion than the central ones,while player 2 is the most collusion averse player.

Does the “V” shape hold for any number of players and anyspecification of the preferences?

Are there any players that are constantly prone or averse to collusion?

EXAMPLE continued

The Shapley value for our example is

φ(vg ) =

Extreme players are more prone to collusion than the central ones,while player 2 is the most collusion averse player.

Does the “V” shape hold for any number of players and anyspecification of the preferences?

Are there any players that are constantly prone or averse to collusion?

A CLOSED FORM FOR THE SHAPLEY VALUE

Theorem

For each agent i ∈ N

φi (vg ) =∑

j∈N\{n}

ψij(bj − bj+1) (2)

where, for each j ∈ N \ {n}

ψij =

{−j c(n, j) if j < i

(n − j) c(n, j) if j ≥ i(3)

and c(n, j) =2n − 3j − j2

2n2(j + 1)(j + 2)

Therefore, the Shapley value can be computed in polynomial time

PROVING THE THEOREMEach value of the coalition in the gain game is a linear combinationbj − bj+1 ⇒ The Shapley value too is a linear combination

Each ψij is the Shapley value for player i in a gain game withvaluations:

{1 if 1 ≤ h ≤ j

0 if j + 1 ≤ h ≤ n

The Theorem requires the following

COMBINATORIAL LEMMA

For every j , t ∈ N,

t∑s=1

)(j+ts

j + 1; (4)

t∑s=1

)(j+ts

) =t (j + t + 1)

(j + 1)(j + 2). (5)

{1 if 1 ≤ h ≤ j

0 if j + 1 ≤ h ≤ n

COMBINATORIAL LEMMA

t∑s=1

)(j+ts

j + 1; (4)

t∑s=1

)(j+ts

) =t (j + t + 1)

(j + 1)(j + 2). (5)

{1 if 1 ≤ h ≤ j

0 if j + 1 ≤ h ≤ n

COMBINATORIAL LEMMA

t∑s=1

)(j+ts

j + 1; (4)

t∑s=1

)(j+ts

) =t (j + t + 1)

(j + 1)(j + 2). (5)

EXAMPLE continuedThe coefficients ψij for a division with 4 players

Players/diff 240-192 192-80 80-48 Shapley

1 1/16 -1/96 -1/64 4/3

2 -1/48 -1/96 -1/64 -8/3

3 -1/48 1/96 -1/64 -1/3

4 -1/48 1/96 3/64 5/3

If we only consider the signs, the Shapley value for player 2 will always benegative, independently of the evaluations

Players/diff b1 − b2 b2 − b3 b3 − b4 Shapley

1 + − − +/−2 − − − −3 − + − +/−4 − + + +/−

The Shapley value will always be “V” shaped with minimum value for Pl.2

EXAMPLE continuedThe coefficients ψij for a division with 4 players

Players/diff 240-192 192-80 80-48 Shapley

1 1/16 -1/96 -1/64 4/3

2 -1/48 -1/96 -1/64 -8/3

3 -1/48 1/96 -1/64 -1/3

4 -1/48 1/96 3/64 5/3

If we only consider the signs, the Shapley value for player 2 will always benegative, independently of the evaluations

1 + − − +/−2 − − − −3 − + − +/−4 − + + +/−

ATTITUDES TOWARD COLLUSION

DEFINITION

Player i is

Strongly (Weakly) collusion prone w.r.t. any specification of theevaluations if ψij > 0 (ψij ≥ 0) for every j

Strongly (Weakly) collusion averse w.r.t. any specification of theevaluations if ψij < 0 (ψij ≤ 0) for every j

The previous example showed that, when n = 4, player 2 is stronglycollusion averse, while there is no agent which is collusion prone.

1 + − − +/−2 − − − −3 − + − +/−4 − + + +/−

ATTITUDES TOWARD COLLUSION

DEFINITION

Player i is

Strongly (Weakly) collusion prone w.r.t. any specification of theevaluations if ψij > 0 (ψij ≥ 0) for every j

Strongly (Weakly) collusion averse w.r.t. any specification of theevaluations if ψij < 0 (ψij ≤ 0) for every j

The previous example showed that, when n = 4, player 2 is stronglycollusion averse, while there is no agent which is collusion prone.

1 + − − +/−2 − − − −3 − + − +/−4 − + + +/−

THE FIRST 5 CASESSign of ψij : “+”, “−” and “0” indicate, respectively, a positive, negativeor null ψij .

[1 02 0

] + −2 − −− +

+ − −2 − − −− + −− + +

n = 2 n = 3 n=4

+ 0 − −

2 − 0 − −3 − 0 − −− 0 + −− 0 + +

+ + − − −− + − − −

3 − − − − −− − + − −− − + + −− − + + +

n = 5 n = 6

THE FIRST 5 CASESSign of ψij : “+”, “−” and “0” indicate, respectively, a positive, negativeor null ψij .

[1 02 0

] + −2 − −− +

+ − −2 − − −− + −− + +

n = 2 n = 3 n=4

+ 0 − −

2 − 0 − −3 − 0 − −− 0 + −− 0 + +

+ + − − −− + − − −

3 − − − − −− − + − −− − + + −− − + + +

n = 5 n = 6

THE FOLLOWING PATTERN HOLDS

Ag 1 2 3 4 5 6 7 8 9 10 11 12 13 14

2 F F3 F F F4 F F F F5 F F F F F6 F F F F F F7 F F F F F F F8 F F F F F F F F9 F F F F F F F F F

10 F F F F F F F F F F11 F F F F F F F F F F F12 F F F F F F F F F F F F13 F F F F F F F F F F F F F14 F F F F F F F F F F F F F F15 F F F F F F F F F F F F F · · ·F = st. averse F = w. averse F = neither prone nor averseM.Dall’Aglio (LUISS) Collusion and Knaster Campione D’Italia 2018 16 / 25

THE GENERAL “COLLUSION PATTERN”

THEOREM

(a) Suppose

h2 + h − 2

2< n <

h2 + 3h

2for some integer h < n

then player h is the only strongly collusion averse one.Moreover φi is decreasing up to h and increasing after it.

(b) If, instead

n =h2 + h − 2

2for some integer h < n

then players h − 1 and h are the only weakly collusion averse players.Moreover φi is decreasing up to h − 1 and increasing after h.

(c) For n > 2 there is no collusion prone player

THE MATRIX {ψij}: ONE S. AVERSE PLAYER

Case 1

h2 + h − 2

2< n <

h2 + 3h + 1

2=⇒ gap changes sign at j = h

Pls/ψij 1 h−1 h n

1 + · · · + − · · · −... · · ·

...... · · ·

...h−1 − · · · + − · · · −h − · · · − − · · · −

h+1 − · · · − + · · · −... · · ·

...... · · ·

...n − · · · − + · · · +

Sign of ψij : “+” and “−” indicate, respectively, a positive or negative ψij .

THE MATRIX {ψij}: TWO W. AVERSE PLAYERS

Case 2

n =h2 + h − 2

2=⇒ ψjj − ψj+1,j = 0 when j = h − 1

Pls/ψij 1 h−1 n

1 + · · · + 0 − · · · −... · · ·

......

... · · ·...

− · · · + 0 − · · · −h−1 − · · · − 0 − · · · −h − · · · − 0 − · · · −

− · · · − 0 + · · · −... · · ·

......

... · · ·...

n − · · · − 0 + · · · +

Sign of ψij : “+”, “−” and “0” indicate, respectively, a positive, negativeor null ψij .

COALITIONS WITH MAXIMAL GAIN

A coalition of s players has maximal gain if it contains the first and thelast s − 1 players

We consider the largest s∗ that achieves the maximum

vg (Ss∗) = maxs=2,3,...,n

vg (Ss) > vg (Ss∗+1)

and the largest s∗∗ that achieves the maximum per capita gain

vg (Ss∗∗) = maxs=2,3,...,n

vg (Ss)

vg (Ss∗∗+1)

s∗∗ + 1

SOME PLAYERS WILL NEVER ENTER MAXIMALCOALITIONS

Theorem

If b1 6= bn then

s∗∗ ≤ s∗

2 ≤ s∗ ≤⌊n

2+ 1⌋

2 ≤ s∗∗ ≤⌈√

and the bounds are sharp.

Players {2, 3, . . . ,⌈n−12

⌉} will never join the coaltion with maximal

(absolute) gain

Players {2, 3, . . . ,⌈√

n⌉− 1} will never join the coaltion with maximal

per capita gain

THE BOUNDS ARE SHARP

Examples

when b1 = b2 = · · · = bn−1 = 1 and bn = 0 then s∗ = s∗∗ = 2

when b1 = 1 and b2 = · · · = bn−1 = 0 then

s∗ =⌊n

2+ 1⌋

s∗∗⌈√

A WORD ON COALITION FORMATION

We consider the increment ∆(s) = vg (Ss)− vg (Ss−1)

Proposition

For any s = 2, 3, . . . , n,

∆(s) is non increasing in s

If b1 6= bn, then ∆(2) > 0

If b2 6= bn, then ∆(n) < 0

If players know their rank (but not the values bj of the other players:

Coalition S2 = {1, n} will form first

Sequentially, players n − 1, n − 2, . . . , s, . . ., will enter the coalition aslong as ∆(s) > 0

AN EARLIER WORK(Graham, Marshall, Richards, 1990)

N = {1, 2, . . . , n} set of players;

b1 ≥ b2 ≥ · · · ≥ bn valuations of item from the players

Players are interested in a second price auction on the item.

A group of them S ⊆ N, called ring, colludes beforehand.

Prior to the auctions, all members of the ring reveal their valuationamong themselves

A a single representative is sent to the auction to bidvmax(S) = maxi∈S bi

N = {1, 2, . . . , n} set of players;

AN EARLIER WORKHow to share the gain of the coalition?

The gain of coalition S is

vAU(S) = max{vmax(S)− vmax(Sc), 0} = b1 −maxj∈Sc

The author suggest a way of dividing the value b1 of the item amongthe players, as the Shapley value of this game

n∑k=j

bk − bk−1k

assuming v0 = 0

This game has the same structure of the polluted river game

n∑k=j

bk − bk−1k

assuming v0 = 0

n∑k=j

bk − bk−1k

assuming v0 = 0

the shapley value in knaster's collusion game€¦ · the shapley value in knaster’s...

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