Blotto, Lotto, All Pay! - ma.huji.ac.ilhart/papers/blotto-p.pdf · Papers Sergiu Hart, Discrete Colonel Blotto and General Lotto Games Int J Game Theory 2008 SERGIU HART °c 2015

Blotto, Lotto, ... All Pay!

Sergiu Hart

June 2017

SERGIU HART c© 2015 – p. 1


Sergiu HartCenter for the Study of Rationality

Dept of Mathematics Dept of EconomicsThe Hebrew University of Jerusalem

[email protected]://www.ma.huji.ac.il/hart


http://www.ma.huji.ac.il/hart

Papers


http://www.ma.huji.ac.il/hart/abs/blotto.html

http://www.ma.huji.ac.il/hart/abs/lotto.html

http://www.ratio.huji.ac.il/sites/default/files/publications/dp687.pdf

Papers

Sergiu Hart, Discrete Colonel Blotto andGeneral Lotto GamesInt J Game Theory 2008www.ma.huji.ac.il/hart/abs/blotto.html





Papers


Sergiu Hart, Allocation Games with Caps:From Captain Lotto to All-Pay AuctionsInt J Game Theory 2016www.ma.huji.ac.il/hart/abs/lotto.html





Papers


Sergiu Hart, Allocation Games with Caps:From Captain Lotto to All-Pay AuctionsInt J Game Theory 2016www.ma.huji.ac.il/hart/abs/lotto.html

Nadav Amir, Uniqueness of OptimalStrategies in Captain Lotto GamesInt J Game Theory 2017/8www.ratio.huji.ac.il/sites/default/files/publications/dp687.pdf





Colonel Blotto Games



Player A has A aquamarine marbles




Player B has B blue marbles





The players distribute their marblesinto K distinct urns






Each urn is CAPTURED by the player who putmore marbles in it







The player who captures more urns WINS thegame








(two-person zero-sum game:win = 1, lose = −1, tie = 0)








Borel 1921SERGIU HART c© 2015 – p. 4







Borel 1921. . .




Colonel Lotto Games


Colonel Lotto Games




Colonel Lotto Games



The players distribute their marblesinto K indistinguishable urns


Colonel Lotto Games




One urn is selected at random (uniformly)


Colonel Lotto Games





The player who put more marbles in theselected urn WINS the game


Colonel Lotto Games





The player who put more marbles in theselected urn WINS the game

Colonel Blotto and Colonel Lotto games areequivalent (same value, same optimalstrategies modulo symmetrization)


Colonel Lotto Games


Colonel Lotto Games

The number of aquamarine marbles in theselected urn is a random variable X ≥ 0 withexpectation a = A/K and integer values


Colonel Lotto Games


The number of blue marbles in the selectedurn is a random variable Y ≥ 0 withexpectation b = B/K and integer values


Colonel Lotto Games



Payoff function:

H(X, Y ) = P[X > Y ] − P[X < Y ]


Colonel Lotto Games



Payoff function:

H(X, Y ) = P[X > Y ] − P[X < Y ]

(X and Y are independent)


General Lotto Games

Player A chooses (the distribution of) arandom variable X ≥ 0 with expectation aand integer values

Player B chooses (the distribution of) arandom variable Y ≥ 0 with expectation band integer values

Payoff function:

H(X, Y ) = P[X > Y ] − P[X < Y ]



Continuous General Lotto Games



Payoff function:

H(X, Y ) = P[X > Y ] − P[X < Y ]






Theorem Let a = b > 0.




VALUE = 0




VALUE = 0

The unique OPTIMAL STRATEGIES :X∗ = Y ∗ = UNIFORM(0, 2a)




VALUE = 0

The unique OPTIMAL STRATEGIES :X∗ = Y ∗ = UNIFORM(0, 2a)

Bell & Cover 1980, Myerson 1993, Lizzeri 1999SERGIU HART c© 2015 – p. 9




Theorem Let a ≥ b > 0.




VALUE = a − ba = 1 − b

a





a

The unique OPTIMAL STRATEGY of A :

X∗ = UNIFORM(0, 2a)(

1 − ba

)





a



1 − ba

)

The unique OPTIMAL STRATEGY of B :

Y ∗ =(

1 − ba

)

10 + ba UNIFORM(0, 2a)





a



1 − ba

)

The unique OPTIMAL STRATEGY of B :

Y ∗ =(

1 − ba

)


Sahuguet & Persico 2006, Hart 2008SERGIU HART c© 2015 – p. 10




Proof. Let U = UNIF(0, 2a).




For every Y ≥ 0 with E[Y ] = b :

P[Y ≥ U ]





P[Y ≥ U ]

=

∫ 2a

0

P[Y ≥ x]1

2adx





P[Y ≥ U ]

=

∫ 2a

0

P[Y ≥ x]1

2adx

≤1

2a

∫ ∞

0

P[Y ≥ x] dx





P[Y ≥ U ]

=

∫ 2a

0

P[Y ≥ x]1

2adx

≤1

2a

∫ ∞

0

P[Y ≥ x] dx

=1

2aE[Y ] =

b

a





P[Y ≥ U ] ≤b

2a





P[Y ≥ U ] ≤b

2a⇒

H(U, Y )





P[Y ≥ U ] ≤b

2a⇒

H(U, Y ) = P[U > Y ] − P[Y > U ]





P[Y ≥ U ] ≤b

2a⇒

H(U, Y ) = P[U > Y ] − P[Y > U ]

≥ 1 − 2P[Y ≥ U ]





P[Y ≥ U ] ≤b

2a⇒

H(U, Y ) = P[U > Y ] − P[Y > U ]

≥ 1 − 2P[Y ≥ U ] ≥ 1 −b

a





H(U, Y ) ≥ 1 −b

a





H(U, Y ) ≥ 1 −b

a

For every X ≥ 0 with E[X] = a :





H(U, Y ) ≥ 1 −b

a


H

(

X,

(

1 −b

a

)

10 +b

aU

)





H(U, Y ) ≥ 1 −b

a


H

(

X,

(

1 −b

a

)

10 +b

aU

)

≤

(

1 −b

a

)

· 1 +b

a· H(X, U)





H(U, Y ) ≥ 1 −b

a


H

(

X,

(

1 −b

a

)

10 +b

aU

)

≤

(

1 −b

a

)

· 1 +b

a· 0





H(U, Y ) ≥ 1 −b

a


H

(

X,

(

1 −b

a

)

10 +b

aU

)

≤

(

1 −b

a

)

· 1 +b

a· 0 = 1 −

b

a




a

OPTIMAL STRATEGY of A :X∗ = UNIFORM(0, 2a)

(

1 − ba

)

OPTIMAL STRATEGY of B :Y ∗ =

(

1 − ba

)





a


(

1 − ba

)


(

1 − ba

)


Uniqueness of OPTIMAL STRATEGIES:




a


(

1 − ba

)


(

1 − ba

)


Uniqueness of OPTIMAL STRATEGIES:express X as an average of two-pointdistributions. . .







Payoff function:

H(X, Y ) = P[X > Y ] − P[X < Y ]



(Discrete) General Lotto Games



Payoff function:

H(X, Y ) = P[X > Y ] − P[X < Y ]






Theorem



Theorem

For integer a ≥ b ≥ 0:

VALUE(a, b) = a − ba



Theorem

For integer a ≥ b ≥ 0:

VALUE(a, b) = a − ba

For general a, b ≥ 0:

interpolate linearly between consecutiveintegers (separately on a and b)



Theorem (continued)



Theorem (continued)

OPTIMAL STRATEGIES:



Theorem (continued)

OPTIMAL STRATEGIES:

UNIF(0, 2a) for integer a is replaced by



Theorem (continued)

OPTIMAL STRATEGIES:


UNIF{0, 2, 4, ..., 2a}



Theorem (continued)

OPTIMAL STRATEGIES:


UNIF{0, 2, 4, ..., 2a}or

UNIF{1, 3, 5, ..., 2a − 1}



Theorem (continued)

OPTIMAL STRATEGIES:


UNIF{0, 2, 4, ..., 2a}or

UNIF{1, 3, 5, ..., 2a − 1}or

averages of the above



Theorem (continued)

OPTIMAL STRATEGIES:


UNIF{0, 2, 4, ..., 2a} (uniform on evens)or

UNIF{1, 3, 5, ..., 2a − 1}or




Theorem (continued)

OPTIMAL STRATEGIES:


UNIF{0, 2, 4, ..., 2a} (uniform on evens)or

UNIF{1, 3, 5, ..., 2a − 1} (uniform on odds)or



Colonel B/Lotto Games



IMPLEMENT the optimal strategiesof the corresponding General Lotto game

by RANDOM PARTITIONS





(explicit constructions are provided)

Hart 2008SERGIU HART c© 2015 – p. 17




(explicit constructions are provided)

Hart 2008, Dziubinski 2013SERGIU HART c© 2015 – p. 17



Colonel B/Lotto Games: Example



A = 7, K = 3



A = 7, K = 3 ⇒ a = A/K = 2 1/3



A = 7, K = 3 ⇒ a = A/K = 2 1/3

Continuous General Lotto:UNIF(0, 2a) = UNIF(0, 4 2/3)



A = 7, K = 3 ⇒ a = A/K = 2 1/3


Discrete General Lotto:UNIF{0, 2, 4} with probability 2/3

UNIF{1, 3, 5} with probability 1/3



A = 7, K = 3 ⇒ a = A/K = 2 1/3


Discrete General Lotto:UNIF{0, 2, 4} with probability 2/3

UNIF{1, 3, 5} with probability 1/3

Discrete Colonel B/Lotto:4 + 2 + 1 = 7 with probability 1/3

4 + 3 + 0 = 7 with probability 1/3

5 + 2 + 0 = 7 with probability 1/3


All-Pay Auction


All-Pay Auction

Object: worth vA to Player Aworth vB to Player B


All-Pay Auction


Player A bids XPlayer B bids Y


All-Pay Auction



Highest bidder wins the object


All-Pay Auction



Highest bidder wins the object

BOTH players pay their bids


All-Pay: The Expenditure Game



[E1] The players choose their EXPECTED BIDS("expenditure"): a = E[X], b = E[Y ]




[E2] Given a and b, the players choose their bidsX and Y (with E[X] = a, E[Y ] = b) so asto maximize the probability of winning





[E2] = General Lotto game





[E2] = General Lotto game⇒ value (and optimal strategies) already solved





[E2] = General Lotto game⇒ value (and optimal strategies) already solved⇒ substitute in [E1]





[E2] = General Lotto game⇒ value (and optimal strategies) already solved⇒ substitute in [E1]⇒ [E1] = simple game on a rectangle





[E2] = General Lotto game⇒ value (and optimal strategies) already solved⇒ substitute in [E1]⇒ [E1] = simple game on a rectangle

→ find Nash equilibria of [E1]





[E2] = General Lotto game⇒ value (and optimal strategies) already solved⇒ substitute in [E1]⇒ [E1] = simple concave game on a rectangle

→ find pure Nash equilibria of [E1]





[E2] = General Lotto game⇒ value (and optimal strategies) already solved⇒ substitute in [E1]⇒ [E1] = simple concave game on a rectangle

→ find pure Nash equilibria of [E1]


[E1]: Best Replies


[E1]: Best Replies

a

b

vA = vB = v


[E1]: Best Replies

a

b

v2

vA = vB = v


[E1]: Best Replies

a

b

v2

v2

vA = vB = v


[E1]: Best Replies and Equilibrium

a

b

v2

v2

vA = vB = v



a

b

v2

v2

a

b

vB

2

vB

2

vA

2

vA

2

vA = vB = v vA > vB



a

b

v2

v2

a

b

vB

2

vB

2

vA

2

vA

2

vA = vB = v vA > vB


General Lotto Games


General Lotto Games

Player A chooses (the distribution of) arandom variable X ≥ 0 with expectation a

Player B chooses (the distribution of) arandom variable Y ≥ 0 with expectation b

Payoff function:

H(X, Y ) = P[X > Y ] − P[X < Y ]



General Lotto Games

Player A chooses (the distribution of) arandom variable X ≥ 0 with expectation aand upper bound (CAP) cA: X ≤ cA

Player B chooses (the distribution of) arandom variable Y ≥ 0 with expectation band upper bound (CAP) cB: Y ≤ cB

Payoff function:

H(X, Y ) = P[X > Y ] − P[X < Y ]



Captain Lotto Games

Player A chooses (the distribution of) arandom variable X ≥ 0 with expectation aand upper bound (CAP) cA: X ≤ cA

Player B chooses (the distribution of) arandom variable Y ≥ 0 with expectation band upper bound (CAP) cB: Y ≤ cB

Payoff function:

H(X, Y ) = P[X > Y ] − P[X < Y ]



Captain Lotto Games


Captain Lotto Games

Theorem Let cA = cB = c ≥ a ≥ b ≥ 0. Then


Captain Lotto Games



a


Captain Lotto Games



a

OPTIMAL STRATEGY X∗ of A :

UNIF(0, 2a) if c ≥ 2a(

ca

− 1)

UNIF(0, 2a)+(

2 − ca

)

1c if c ≤ 2a

OPTIMAL STRATEGY Y ∗ of B :(

1 − ba

)

10 + ba

X∗


Captain Lotto Games



a

OPTIMAL STRATEGY X∗ of A :


ca

− 1)

UNIF(0, 2a)+(

2 − ca

)

1c if c ≤ 2a

OPTIMAL STRATEGY Y ∗ of B :(

1 − ba

)

10 + ba

X∗


Captain Lotto Games



a

Unique OPTIMAL STRATEGY X∗ of A :


ca

− 1)

UNIF(0, 2a)+(

2 − ca

)

1c if c ≤ 2a

Unique OPTIMAL STRATEGY Y ∗ of B :(

1 − ba

)

10 + ba

X∗


Captain Lotto Games



a

Unique OPTIMAL STRATEGY X∗ of A :


ca

− 1)

UNIF(0, 2a)+(

2 − ca

)

1c if c ≤ 2a

Unique OPTIMAL STRATEGY Y ∗ of B :(

1 − ba

)

10 + ba

X∗

Hart 2016, Amir 2017SERGIU HART c© 2015 – p. 23

Optimal Strategy X∗



c2a

12a

2a ≤ c



c2a

12a

2a ≤ c

2c − a c 2a

12a

c ≤ 2a


Captain Lotto: Unequal Caps



Theorem Let cA > cB = c > 0, and let0 ≤ a ≤ cA and 0 ≤ b ≤ c. Then the VALUE is:

a−bmax{a,b}

if 0 < max{a, b} ≤ c2

1 − 4b(c−a)c2

if b ≤ c2

and c2

≤ a ≤ c

2ac

− 1 if c2

≤ b ≤ c and a ≤ c

1 if a ≥ c



Theorem Let cA > cB = c > 0, and let0 ≤ a ≤ cA and 0 ≤ b ≤ c. Then the VALUE is:

a−bmax{a,b}

if 0 < max{a, b} ≤ c2

1 − 4b(c−a)c2

if b ≤ c2

and c2

≤ a ≤ c

2ac

− 1 if c2

≤ b ≤ c and a ≤ c

1 if a ≥ c

The (unique) OPTIMAL STRATEGIES X∗, Y ∗ are...


All-Pay Auctions with Caps



Applying the results on Captain LottoGames to the Expenditure Games



Applying the results on Captain LottoGames to the Expenditure Gamessolves All-Pay Auctions with Caps



Applying the results on Captain LottoGames to the Expenditure Gamessolves All-Pay Auctions with Caps• New results: unequal caps







Theorem. Let vA ≥ vB > 0. The maximalrevenue from all-pay auctions with possiblecaps




Theorem. Let vA ≥ vB > 0. The maximalrevenue from all-pay auctions with possiblecaps is obtained when the bid X of Player Ais capped at cA = vB − ε and the bid Y ofPlayer B is uncapped.




Theorem. Let vA ≥ vB > 0. The maximalrevenue from all-pay auctions with possiblecaps is obtained when the bid X of Player Ais capped at cA = vB − ε and the bid Y ofPlayer B is uncapped.

Hart 2016, Szech 2015SERGIU HART c© 2015 – p. 26





Blotto, Lotto, All Pay! - ma.huji.ac.ilhart/papers/blotto-p.pdf · Papers Sergiu Hart, Discrete Colonel Blotto and General Lotto Games Int J Game Theory 2008 SERGIU HART °c 2015

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