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Blotto, Lotto, ... All Pay!
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
SERGIU HART c© 2015 – p. 2
Papers
SERGIU HART c© 2015 – p. 3
Papers
Sergiu Hart, Discrete Colonel Blotto andGeneral Lotto GamesInt J Game Theory 2008www.ma.huji.ac.il/hart/abs/blotto.html
SERGIU HART c© 2015 – p. 3
Papers
Sergiu Hart, Discrete Colonel Blotto andGeneral Lotto GamesInt J Game Theory 2008www.ma.huji.ac.il/hart/abs/blotto.html
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
SERGIU HART c© 2015 – p. 3
Papers
Sergiu Hart, Discrete Colonel Blotto andGeneral Lotto GamesInt J Game Theory 2008www.ma.huji.ac.il/hart/abs/blotto.html
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
SERGIU HART c© 2015 – p. 3
Colonel Blotto Games
Player A has A aquamarine marbles
Player B has B blue marbles
SERGIU HART c© 2015 – p. 4
Colonel Blotto Games
Player A has A aquamarine marbles
Player B has B blue marbles
The players distribute their marblesinto K distinct urns
SERGIU HART c© 2015 – p. 4
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
SERGIU HART c© 2015 – p. 4
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
SERGIU HART c© 2015 – p. 4
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)
SERGIU HART c© 2015 – p. 4
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
Borel 1921SERGIU HART c© 2015 – p. 4
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
Borel 1921. . .
SERGIU HART c© 2015 – p. 4
Colonel Lotto Games
Player A has A aquamarine marbles
Player B has B blue marbles
SERGIU HART c© 2015 – p. 5
Colonel Lotto Games
Player A has A aquamarine marbles
Player B has B blue marbles
The players distribute their marblesinto K indistinguishable urns
SERGIU HART c© 2015 – p. 5
Colonel Lotto Games
Player A has A aquamarine marbles
Player B has B blue marbles
The players distribute their marblesinto K indistinguishable urns
One urn is selected at random (uniformly)
SERGIU HART c© 2015 – p. 5
Colonel Lotto Games
Player A has A aquamarine marbles
Player B has B blue marbles
The players distribute their marblesinto K indistinguishable urns
One urn is selected at random (uniformly)
The player who put more marbles in theselected urn WINS the game
SERGIU HART c© 2015 – p. 5
Colonel Lotto Games
Player A has A aquamarine marbles
Player B has B blue marbles
The players distribute their marblesinto K indistinguishable urns
One urn is selected at random (uniformly)
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)
SERGIU HART c© 2015 – p. 5
Colonel Lotto Games
The number of aquamarine marbles in theselected urn is a random variable X ≥ 0 withexpectation a = A/K and integer values
SERGIU HART c© 2015 – p. 6
Colonel Lotto Games
The number of aquamarine marbles in theselected urn is a random variable X ≥ 0 withexpectation a = A/K and integer values
The number of blue marbles in the selectedurn is a random variable Y ≥ 0 withexpectation b = B/K and integer values
SERGIU HART c© 2015 – p. 6
Colonel Lotto Games
The number of aquamarine marbles in theselected urn is a random variable X ≥ 0 withexpectation a = A/K and integer values
The number of blue marbles in the selectedurn is a random variable Y ≥ 0 withexpectation b = B/K and integer values
Payoff function:
H(X, Y ) = P[X > Y ] − P[X < Y ]
SERGIU HART c© 2015 – p. 6
Colonel Lotto Games
The number of aquamarine marbles in theselected urn is a random variable X ≥ 0 withexpectation a = A/K and integer values
The number of blue marbles in the selectedurn is a random variable Y ≥ 0 withexpectation b = B/K and integer values
Payoff function:
H(X, Y ) = P[X > Y ] − P[X < Y ]
(X and Y are independent)
SERGIU HART c© 2015 – p. 6
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 ]
(X and Y are independent)
SERGIU HART c© 2015 – p. 7
Continuous 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 ]
(X and Y are independent)
SERGIU HART c© 2015 – p. 8
Continuous General Lotto Games
Theorem Let a = b > 0.
VALUE = 0
The unique OPTIMAL STRATEGIES :X∗ = Y ∗ = UNIFORM(0, 2a)
SERGIU HART c© 2015 – p. 9
Continuous General Lotto Games
Theorem Let a = b > 0.
VALUE = 0
The unique OPTIMAL STRATEGIES :X∗ = Y ∗ = UNIFORM(0, 2a)
Bell & Cover 1980, Myerson 1993, Lizzeri 1999SERGIU HART c© 2015 – p. 9
Continuous General Lotto Games
Theorem Let a ≥ b > 0.
VALUE = a − ba = 1 − b
a
SERGIU HART c© 2015 – p. 10
Continuous General Lotto Games
Theorem Let a ≥ b > 0.
VALUE = a − ba = 1 − b
a
The unique OPTIMAL STRATEGY of A :
X∗ = UNIFORM(0, 2a)(
1 − ba
)
SERGIU HART c© 2015 – p. 10
Continuous General Lotto Games
Theorem Let a ≥ b > 0.
VALUE = a − ba = 1 − b
a
The unique OPTIMAL STRATEGY of A :
X∗ = UNIFORM(0, 2a)(
1 − ba
)
The unique OPTIMAL STRATEGY of B :
Y ∗ =(
1 − ba
)
10 + ba UNIFORM(0, 2a)
SERGIU HART c© 2015 – p. 10
Continuous General Lotto Games
Theorem Let a ≥ b > 0.
VALUE = a − ba = 1 − b
a
The unique OPTIMAL STRATEGY of A :
X∗ = UNIFORM(0, 2a)(
1 − ba
)
The unique OPTIMAL STRATEGY of B :
Y ∗ =(
1 − ba
)
10 + ba UNIFORM(0, 2a)
Sahuguet & Persico 2006, Hart 2008SERGIU HART c© 2015 – p. 10
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
P[Y ≥ U ]
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
P[Y ≥ U ]
=
∫ 2a
0
P[Y ≥ x]1
2adx
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
P[Y ≥ U ]
=
∫ 2a
0
P[Y ≥ x]1
2adx
≤1
2a
∫ ∞
0
P[Y ≥ x] dx
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
P[Y ≥ U ]
=
∫ 2a
0
P[Y ≥ x]1
2adx
≤1
2a
∫ ∞
0
P[Y ≥ x] dx
=1
2aE[Y ] =
b
a
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
P[Y ≥ U ] ≤b
2a
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
P[Y ≥ U ] ≤b
2a⇒
H(U, Y )
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
P[Y ≥ U ] ≤b
2a⇒
H(U, Y ) = P[U > Y ] − P[Y > U ]
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
P[Y ≥ U ] ≤b
2a⇒
H(U, Y ) = P[U > Y ] − P[Y > U ]
≥ 1 − 2P[Y ≥ U ]
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
P[Y ≥ U ] ≤b
2a⇒
H(U, Y ) = P[U > Y ] − P[Y > U ]
≥ 1 − 2P[Y ≥ U ] ≥ 1 −b
a
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
H(U, Y ) ≥ 1 −b
a
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
H(U, Y ) ≥ 1 −b
a
For every X ≥ 0 with E[X] = a :
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
H(U, Y ) ≥ 1 −b
a
For every X ≥ 0 with E[X] = a :
H
(
X,
(
1 −b
a
)
10 +b
aU
)
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
H(U, Y ) ≥ 1 −b
a
For every X ≥ 0 with E[X] = a :
H
(
X,
(
1 −b
a
)
10 +b
aU
)
≤
(
1 −b
a
)
· 1 +b
a· H(X, U)
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
H(U, Y ) ≥ 1 −b
a
For every X ≥ 0 with E[X] = a :
H
(
X,
(
1 −b
a
)
10 +b
aU
)
≤
(
1 −b
a
)
· 1 +b
a· 0
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
Proof. Let U = UNIF(0, 2a).
For every Y ≥ 0 with E[Y ] = b :
H(U, Y ) ≥ 1 −b
a
For every X ≥ 0 with E[X] = a :
H
(
X,
(
1 −b
a
)
10 +b
aU
)
≤
(
1 −b
a
)
· 1 +b
a· 0 = 1 −
b
a
SERGIU HART c© 2015 – p. 11
Continuous General Lotto Games
VALUE = a − ba = 1 − b
a
OPTIMAL STRATEGY of A :X∗ = UNIFORM(0, 2a)
(
1 − ba
)
OPTIMAL STRATEGY of B :Y ∗ =
(
1 − ba
)
10 + ba UNIFORM(0, 2a)
SERGIU HART c© 2015 – p. 12
Continuous General Lotto Games
VALUE = a − ba = 1 − b
a
OPTIMAL STRATEGY of A :X∗ = UNIFORM(0, 2a)
(
1 − ba
)
OPTIMAL STRATEGY of B :Y ∗ =
(
1 − ba
)
10 + ba UNIFORM(0, 2a)
Uniqueness of OPTIMAL STRATEGIES:
SERGIU HART c© 2015 – p. 12
Continuous General Lotto Games
VALUE = a − ba = 1 − b
a
OPTIMAL STRATEGY of A :X∗ = UNIFORM(0, 2a)
(
1 − ba
)
OPTIMAL STRATEGY of B :Y ∗ =
(
1 − ba
)
10 + ba UNIFORM(0, 2a)
Uniqueness of OPTIMAL STRATEGIES:express X as an average of two-pointdistributions. . .
SERGIU HART c© 2015 – p. 12
Continuous 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 ]
(X and Y are independent)
SERGIU HART c© 2015 – p. 13
(Discrete) 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 ]
(X and Y are independent)
SERGIU HART c© 2015 – p. 14
(Discrete) General Lotto Games
Theorem
For integer a ≥ b ≥ 0:
VALUE(a, b) = a − ba
SERGIU HART c© 2015 – p. 15
(Discrete) General Lotto Games
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)
SERGIU HART c© 2015 – p. 15
(Discrete) General Lotto Games
Theorem (continued)
OPTIMAL STRATEGIES:
UNIF(0, 2a) for integer a is replaced by
SERGIU HART c© 2015 – p. 16
(Discrete) General Lotto Games
Theorem (continued)
OPTIMAL STRATEGIES:
UNIF(0, 2a) for integer a is replaced by
UNIF{0, 2, 4, ..., 2a}
SERGIU HART c© 2015 – p. 16
(Discrete) General Lotto Games
Theorem (continued)
OPTIMAL STRATEGIES:
UNIF(0, 2a) for integer a is replaced by
UNIF{0, 2, 4, ..., 2a}or
UNIF{1, 3, 5, ..., 2a − 1}
SERGIU HART c© 2015 – p. 16
(Discrete) General Lotto Games
Theorem (continued)
OPTIMAL STRATEGIES:
UNIF(0, 2a) for integer a is replaced by
UNIF{0, 2, 4, ..., 2a}or
UNIF{1, 3, 5, ..., 2a − 1}or
averages of the above
SERGIU HART c© 2015 – p. 16
(Discrete) General Lotto Games
Theorem (continued)
OPTIMAL STRATEGIES:
UNIF(0, 2a) for integer a is replaced by
UNIF{0, 2, 4, ..., 2a} (uniform on evens)or
UNIF{1, 3, 5, ..., 2a − 1}or
averages of the above
SERGIU HART c© 2015 – p. 16
(Discrete) General Lotto Games
Theorem (continued)
OPTIMAL STRATEGIES:
UNIF(0, 2a) for integer a is replaced by
UNIF{0, 2, 4, ..., 2a} (uniform on evens)or
UNIF{1, 3, 5, ..., 2a − 1} (uniform on odds)or
averages of the above
SERGIU HART c© 2015 – p. 16
Colonel B/Lotto Games
IMPLEMENT the optimal strategiesof the corresponding General Lotto game
by RANDOM PARTITIONS
SERGIU HART c© 2015 – p. 17
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
Colonel B/Lotto Games
IMPLEMENT the optimal strategiesof the corresponding General Lotto game
by RANDOM PARTITIONS
(explicit constructions are provided)
Hart 2008, Dziubinski 2013SERGIU HART c© 2015 – p. 17
Colonel B/Lotto Games: Example
A = 7, K = 3 ⇒ a = A/K = 2 1/3
Continuous General Lotto:UNIF(0, 2a) = UNIF(0, 4 2/3)
SERGIU HART c© 2015 – p. 18
Colonel B/Lotto Games: Example
A = 7, K = 3 ⇒ a = A/K = 2 1/3
Continuous General Lotto:UNIF(0, 2a) = UNIF(0, 4 2/3)
Discrete General Lotto:UNIF{0, 2, 4} with probability 2/3
UNIF{1, 3, 5} with probability 1/3
SERGIU HART c© 2015 – p. 18
Colonel B/Lotto Games: Example
A = 7, K = 3 ⇒ a = A/K = 2 1/3
Continuous General Lotto:UNIF(0, 2a) = UNIF(0, 4 2/3)
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
SERGIU HART c© 2015 – p. 18
All-Pay Auction
Object: worth vA to Player Aworth vB to Player B
Player A bids XPlayer B bids Y
SERGIU HART c© 2015 – p. 19
All-Pay Auction
Object: worth vA to Player Aworth vB to Player B
Player A bids XPlayer B bids Y
Highest bidder wins the object
SERGIU HART c© 2015 – p. 19
All-Pay Auction
Object: worth vA to Player Aworth vB to Player B
Player A bids XPlayer B bids Y
Highest bidder wins the object
BOTH players pay their bids
SERGIU HART c© 2015 – p. 19
All-Pay: The Expenditure Game
[E1] The players choose their EXPECTED BIDS("expenditure"): a = E[X], b = E[Y ]
SERGIU HART c© 2015 – p. 20
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
SERGIU HART c© 2015 – p. 20
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
SERGIU HART c© 2015 – p. 20
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⇒ value (and optimal strategies) already solved
SERGIU HART c© 2015 – p. 20
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⇒ value (and optimal strategies) already solved⇒ substitute in [E1]
SERGIU HART c© 2015 – p. 20
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⇒ value (and optimal strategies) already solved⇒ substitute in [E1]⇒ [E1] = simple game on a rectangle
SERGIU HART c© 2015 – p. 20
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⇒ value (and optimal strategies) already solved⇒ substitute in [E1]⇒ [E1] = simple game on a rectangle
→ find Nash equilibria of [E1]
SERGIU HART c© 2015 – p. 20
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⇒ value (and optimal strategies) already solved⇒ substitute in [E1]⇒ [E1] = simple concave game on a rectangle
→ find pure Nash equilibria of [E1]
SERGIU HART c© 2015 – p. 20
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⇒ value (and optimal strategies) already solved⇒ substitute in [E1]⇒ [E1] = simple concave game on a rectangle
→ find pure Nash equilibria of [E1]
Hart 2016SERGIU HART c© 2015 – p. 20
[E1]: Best Replies and Equilibrium
a
b
v2
v2
a
b
vB
2
vB
2
vA
2
vA
2
vA = vB = v vA > vB
SERGIU HART c© 2015 – p. 21
[E1]: Best Replies and Equilibrium
a
b
v2
v2
a
b
vB
2
vB
2
vA
2
vA
2
vA = vB = v vA > vB
SERGIU HART c© 2015 – p. 21
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 ]
(X and Y are independent)
SERGIU HART c© 2015 – p. 22
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 ]
(X and Y are independent)
SERGIU HART c© 2015 – p. 22
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 ]
(X and Y are independent)
SERGIU HART c© 2015 – p. 22
Captain Lotto Games
Theorem Let cA = cB = c ≥ a ≥ b ≥ 0. Then
VALUE = a − ba = 1 − b
a
SERGIU HART c© 2015 – p. 23
Captain Lotto Games
Theorem Let cA = cB = c ≥ a ≥ b ≥ 0. Then
VALUE = a − ba = 1 − b
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∗
SERGIU HART c© 2015 – p. 23
Captain Lotto Games
Theorem Let cA = cB = c ≥ a ≥ b ≥ 0. Then
VALUE = a − ba = 1 − b
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∗
Hart 2016SERGIU HART c© 2015 – p. 23
Captain Lotto Games
Theorem Let cA = cB = c ≥ a ≥ b ≥ 0. Then
VALUE = a − ba = 1 − b
a
Unique OPTIMAL STRATEGY X∗ of A :
UNIF(0, 2a) if c ≥ 2a(
ca
− 1)
UNIF(0, 2a)+(
2 − ca
)
1c if c ≤ 2a
Unique OPTIMAL STRATEGY Y ∗ of B :(
1 − ba
)
10 + ba
X∗
Hart 2016SERGIU HART c© 2015 – p. 23
Captain Lotto Games
Theorem Let cA = cB = c ≥ a ≥ b ≥ 0. Then
VALUE = a − ba = 1 − b
a
Unique OPTIMAL STRATEGY X∗ of A :
UNIF(0, 2a) if c ≥ 2a(
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
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
SERGIU HART c© 2015 – p. 25
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
The (unique) OPTIMAL STRATEGIES X∗, Y ∗ are...
SERGIU HART c© 2015 – p. 25
All-Pay Auctions with Caps
Applying the results on Captain LottoGames to the Expenditure Games
SERGIU HART c© 2015 – p. 26
All-Pay Auctions with Caps
Applying the results on Captain LottoGames to the Expenditure Gamessolves All-Pay Auctions with Caps
SERGIU HART c© 2015 – p. 26
All-Pay Auctions with Caps
Applying the results on Captain LottoGames to the Expenditure Gamessolves All-Pay Auctions with Caps• New results: unequal caps
SERGIU HART c© 2015 – p. 26
All-Pay Auctions with Caps
Applying the results on Captain LottoGames to the Expenditure Gamessolves All-Pay Auctions with Caps• New results: unequal caps
Hart 2016SERGIU HART c© 2015 – p. 26
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
Hart 2016SERGIU HART c© 2015 – p. 26
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 is obtained when the bid X of Player Ais capped at cA = vB − ε and the bid Y ofPlayer B is uncapped.
Hart 2016SERGIU HART c© 2015 – p. 26
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 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