Implications of contrarian and
one-sided strategies for the
fair-coin game
Yasunori Horikoshi and Akimichi Takemura
March 31, 2008
Purpose
• Give lower bounds of convergence
• for Strong Law of Large Numbers
• in coin-tossing games
• in game-theoretic probability
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Reference
Implications of contrarian and one-sided strategies
for the fair-coin game. Stochastic Processes and their
Applications,
doi:10.1016/j.spa.2007.11.007.
Y.Horikoshi and A.Takemura.
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Game-theoretic probability
Game-theoretic probability
New framework for probability theory established in
2001 by Shafer and Vovk
“Probability and Finance – It’s Only a Game!”
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Game-theoretic probability
Game
We start with games instead of probability.
Law of large numbers in game-theoretic probability
Game-theoretic statement: there exists a strategy
forcing the law.
Intuition
In the fair game, it is impossible to become
infinitely rich.
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Coin-tossing game (0)
K0 := 1
FOR n = 1, 2, · · ·
Skeptic
announces Mn ∈ R
Reality
announces xn ∈ {−1, 1}
Kn := Kn−1 + Mnxn
END FOR
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⇐ Protocol
Coin tosses are repeated
indefinitely!
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Coin-tossing game (1)
K0 := 1
FOR n = 1, 2, · · ·
Skeptic
announces Mn ∈ R
Reality
announces xn ∈ {−1, 1}
Kn := Kn−1 + Mnxn
END FOR
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Player
Skeptic predicts heads or tails
trying to increase his
capital to infinity
Reality decides heads and
tails, trying to keep
Skeptic’s capital
bounded.
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Coin-tossing game (2)
K0 := 1
FOR n = 1, 2, · · ·
Skeptic
announces Mn ∈ R
Reality
announces xn ∈ {−1, 1}
Kn := Kn−1 + Mnxn
END FOR
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Note:
Realitymoves after Skeptic!
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Coin-tossing game (3)
K0 := 1
FOR n = 1, 2, · · ·
Skeptic
announces Mn ∈ R
Reality
announces xn ∈ {−1, 1}
Kn := Kn−1 + Mnxn
END FOR
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Mn : Skeptic’s bet
xn : result of coin toss
Mnxn : return to Skeptic
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Coin-tossing game (4)
K0 := 1
FOR n = 1, 2, · · ·
Skeptic
announces Mn ∈ R
Reality
announces xn ∈ {−1, 1}
Kn := Kn−1 + Mnxn
END FOR
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Return (1)
Skeptic’s return=Mnxn
Skeptic predicts heads
⇒ Mn > 0
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Coin-tossing game (5)
K0 := 1
FOR n = 1, 2, · · ·
Skeptic
announces Mn ∈ R
Reality
announces xn ∈ {−1, 1}
Kn := Kn−1 + Mnxn
END FOR
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Return (2)
Reality moves after Skeptic!
She can ALWAYS make
Mnxn ≤ 0.
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Coin-tossing game (6)
K0 := 1
FOR n = 1, 2, · · ·
Skeptic
announces Mn ∈ R
Reality
announces xn ∈ {−1, 1}
Kn := Kn−1 + Mnxn
END FOR
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K0 : initial capital of Skeptic
n : number of rounds
Kn : Skeptic’s capital at the
end of round n
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Coin-tossing game (7)
K0 := 1
FOR n = 1, 2, · · ·
Skeptic
announces Mn ∈ R
Reality
announces xn ∈ {−1, 1}
Kn := Kn−1 + Mnxn
END FOR
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Collateral duty
Skeptic loses if he becomes
bankrupt.
Therefore he has to choose
Mn ∈ [−Kn−1, Kn−1].
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Coin-tossing game (8)
K0 := 1
FOR n = 1, 2, · · ·
Skeptic
announces Mn ∈ R
Reality
announces xn ∈ {−1, 1}
Kn := Kn−1 + Mnxn
END FOR
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Notations
s0 := 0
sn := x1 + · · · + xn
(n = 1, 2, . . .)
x0 := 0
sn
xn :=
(n = 1, 2, . . .)
n
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Statement of game-theoretic SLLN (1)
Strategy
Shafer and Vovk proved that there exist a Skeptic’s
strategy such that
whenever Reality does not observe SLLN
(i.e. xn does not converge to 0),
then Skeptic’s capital grows to infinity.
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Statement of game-theoretic SLLN (2)
Meaning of a strategy
Reality can take any of the paths
{x1x2 · · · |xi = 1 or − 1} = {−1, 1}∞.
Whatever path Reality chooses
either Kn → ∞ or xn → 0.
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Reality is faced with only two choices:
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Reality is faced with only two choices:
• Either she observes SLLN,
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Reality is faced with only two choices:
• Either she observes SLLN,
• Or she accepts Skeptic becoming infinitely rich.
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Reality is faced with only two choices:
• Either she observes SLLN,
• Or she accepts Skeptic becoming infinitely rich.
Reality is forced the SLLN!
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Generality of game-theoretic probability
Expectation, probability
Expectation and probability are naturally defined
from the game.
Various probability laws
SLLN for general case, Law of the Iterated
Logarithm, Central Limit Theorem, Black-Scholes
formula, can all be derived.
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Effectiveness of game-theoretic arguments
Simplicity of assumptions
Same statements from fewer assumptions than the
measure-theoretic formulation.
Simplicity of proofs
With clever strategies, proofs are often much
simpler than the measure-theoretic ones.
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Central Limit Theorem and Black-Scholes
formula
• In the BOOK, CLT is proved by a betting strategy.
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Central Limit Theorem and Black-Scholes
formula
• In the BOOK, CLT is proved by a betting strategy.
• Without using characteristic function
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Central Limit Theorem and Black-Scholes
formula
• In the BOOK, CLT is proved by a betting strategy.
• Without using characteristic function
• CLT and Black-Scholes are logically equivalent.
(in game-theoretic probability and hence in
measure-theoretic probability)
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Rate of convergence of SLLN
SLLN
Forcing of |xn| = o(1)
Upper bound for convergence
Kumon and Takemura
(2007)
gave
a
strategy
√
forcing |xn| = O( log n/n)
We studied strategies for lower bounds.
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Rate of convergence of SLLN
SLLN
Forcing of |xn| = o(1)
Upper bound for convergence
Kumon and Takemura
(2007)
gave
a
strategy
√
forcing |xn| = O( log n/n)
We studied strategies for lower bounds.
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Intuition behind momentum strategy (1)
Bet on heads
sn
momentum strategy
sn−1 > 0 ⇒ Bet on heads
n
sn−1 < 0 ⇒ Bet on tails
Bet on tails
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Intuition behind momentum strategy (2)
Kn up
sn
Kn down
Intuition
|sn| % ⇒ Kn %
Kn
n
down
Kn up
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Strategy for lower bound (1)
Strategy Pc
For 0 < c ≤ 1/2
Pc : Mn = −cxn−1Kn−1
This increases the capital when |sn| is small.
(contrarian strategy)
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Intuition behind contrarian strategy (1)
Bet on tails
sn
contrarian strategy
sn−1 > 0 ⇒ Bet on tails
n
sn−1 < 0 ⇒ Bet on heads
Bet on heads
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Intuition behind contrarian strategy (2)
Kn down
sn
Kn up
contrarian strategy
|sn| & ⇒ Kn %
Kn
n
up
Kn down
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Theorem for lower bound (1)
Theorem 1
babababababababababababababababababababababababababababababab
With Pc
√
lim sup(1 + 2c) n|xn| < 1
⇒ lim Kn = ∞.
We can let c ↓ 0. Then this theorem states that
the√rate of convergence is bounded from below by
O( 1/n).
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Theorem for lower bound (2)
Theorem 2
babababababababababababababababababababababababababababababab
Skeptic can force that there are infinite number
of n’s with
√
|sn| ≥ n − 1
√
|sn| ≥ n − 1 for only finite number of n
⇒ Kn → ∞
(stronger than Theorem 1.)
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Strategy for lower bound 2
For Theorem 2 we use the strategy
P ² : Mn = −²sn−1
(again a contrarian strategy.)
However we need a condition for collateral duty.
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Capital process of P ² : Mn = −²sn−1
|sn |
6
5
4
3
2
1
1
±0
0
1
0
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1
2
3
4
5
6
n
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Capital process of P ² : Mn = −²sn−1
|sn |
6
5
4
3
1−
−
2
1
0
1
1
0
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+
1+
1
2
3
4
5
6
n
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Capital process of P ² : Mn = −²sn−1
|sn |
6
5
4
3
2
1
0
1
0
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1 − 3
−2
1−
+2
1+
1
±0
1+
1
2
3
4
5
6
n
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Capital process of P ² : Mn = −²sn−1
|sn |
6
5
4
3
2
1
0
1
0
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1 − 6
−3
1 − 3
+3
1−
1
−
1+
1
+
1 + 2
1+
1
2
3
4
5
6
n
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Capital process of P ² : Mn = −²sn−1
|sn |
1 − 15
6
1 − 10
5
1 − 2
1 − 3
3
1−
2
1
0
1 − 5
1 − 6
4
1+
1
0
1
1 + 2
1+
1
2
1+
1
1 + 2
3
4
1 + 3
5
6
n
Monotone increasing at the same level of |sn|.
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Explicit form of Kn
Generally,
²
Kn = 1 + (n − s2n).
2
Therefore
√
Kn ≥ 0 ⇔ |sn| ≤
n+
2
²
.
and for collateral duty, Skeptic has to stop betting
at the first n such that
√
2
|sn| > n + 1 + − 1.
²
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|sn |
6
5
|sn | =
4
r
n+
2
3
2
1
0
1
2
3
4
5
6
7
8
n
Stop when (n, sn) comes to a red point.
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sn can not be bounded
The capital process of contrarian strategy
²
Kn = 1 + (n − s2n).
2
implies that Skeptic becomes infinitely rich if |sn|
stays bounded.
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Lemma
Lemma 1
babababababababababababababababababababababababababababababab
Skeptic can force that there are infinitely many
n with
sn = 0.
Sketch of proof: Suppose that sn stays nonnegative. We know that sn can not be bounded.
But then a one-sided strategy makes Skeptic
infinitely rich.
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Proof of Theorem 2
¶
³
Proof (We show pictures later)
µ
´
We divide the initial capital of 1 (dollar) into two
accounts A1, A2 with the initial capital of 1/2 each.
We employ the strategy of Lemma 1 for A1 for
forcing sn = 0 infinitely often.
For the account A2 we divide its initial capital of
1/2 into two further accounts A12, A22 (initial capital
of 1/4 each).
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Let L be a large number and let n0 be the first
time after L such that sn = 0.
We keep the initial capital of A22 for later use.
In the account A12, we consider n0 to be the initial
round (i.e. reset the clock) and employ the strategy
²
²0
P , where ²0 := (n0 − 1).
2
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Then we can keep betting for the account A12 until
√
√
2
|sn| > (n − n0) + 1 + − 1 = n − 1. (1)
²
If (1) does not happen after n0, then the amount
of account A12 becomes arbitrarily large.
If (1) happens at some n > n0, we stop betting
and wait until n such that sn = 0. Let this n be
n1. Recall that we have saved the capital of 1/4 in
the account A22. We move 1/8 out of 1/4 into the
account A12. Then we can repeat the procedure!
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Picture for explanation
|sn |
|sn | =
√
n−1
←sample path, jagged actually
L
A12
n
No bet at the beginning
A22
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Picture for explanation
|sn |
|sn | =
L
√
n−1
n
n0
A12
We start when sn = 0
A22
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Picture for explanation
|sn |
|sn | =
L
A12
A22
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n−1
n
n0
A12
√
↓ 0 when the path approaces
√
n−1
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Picture for explanation
|sn |
|sn | =
L
n0
A12
n1
√
n−1
n
We halve A22 and wait until sn = 0.
A22
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Picture for explanation
|sn |
|sn | =
L
n0
n1
A12
If sn does not approch
A22
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√
√
n−1
n
n − 1 thenA12 goes up.
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Capital process of P ² : Mn = −²sn−1
|sn |
1 − 15
6
1 − 10
5
1 − 2
1 − 3
3
1−
2
1
0
1 − 5
1 − 6
4
1+
1
0
1
1 + 2
1+
1
2
1+
1
1 + 2
3
4
1 + 3
5
6
n
Monotone increasing at the same level of |sn|.
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Remark
• contrarian strategy is symmetric w.r.t. the sign.
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Remark
• contrarian strategy is symmetric w.r.t. the sign.
– It can only force properties of |sn|.
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Remark
• contrarian strategy is symmetric w.r.t. the sign.
– It can only force properties of |sn|.
– i.e. Reality can keep sn always positive.
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Remark
• contrarian strategy is symmetric w.r.t. the sign.
– It can only force properties of |sn|.
– i.e. Reality can keep sn always positive.
• We can construct a strategy for preventing this.
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Remark
• contrarian strategy is symmetric w.r.t. the sign.
– It can only force properties of |sn|.
– i.e. Reality can keep sn always positive.
• We can construct a strategy for preventing this.
– Need to force the symmetry between sn and
−sn.
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Remark
• contrarian strategy is symmetric w.r.t. the sign.
– It can only force properties of |sn|.
– i.e. Reality can keep sn always positive.
• We can construct a strategy for preventing this.
– Need to force the symmetry between sn and
−sn.
– Then we can show that our theorem holds for
sn instead of |sn|.
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Summary and concluding remarks
• We gave two strategies for
convergence rate from below.
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bounding
the
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Summary and concluding remarks
• We gave two strategies for
convergence rate from below.
bounding
the
– The idea of contrarian strategy was essential.
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Summary and concluding remarks
• We gave two strategies for
convergence rate from below.
bounding
the
– The idea of contrarian strategy was essential.
• We also showed result on sn instead of |sn|.
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Summary and concluding remarks
• Game-theoretic probability is a very new field. It
offers new ways of looking at probability from the
viewpoint of betting.
• Recently rapid advances have been made
(Vladimir Vovk taking up an approach by our
group) on how to treat continuous-times trading
games.
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