Probability It's only a game!

Probability without Measure!

Mark Saroufim

University of California San Diego

[email protected]

February 18, 2014

Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 1 / 25

Overview

1 History of Probability TheoryBefore KolmogorovDuring KolmogorovAfter Kolmogorov

2 Shafer and VovkIt’s only a gameWinning conditionsComparison with measure theoryAn analogue to variance

3 Efficient Market HypothesisSecurities Market Protocol

A gambler’s perspective

This was the day before probability theory was even a field inmathematics, a field without foundations. Pascal and Fermat simplywanted to win a ton of money betting on horses and wanted to firstsee what it meant for a game to be fair.

If I pay a$You pay a$The winner gets paid 2a$

This is what is referred to as inter alia (equal terms)

P[E ] = how much money you’re willing to put on a game where youcould win 1$

Looking at the real world

Bernoulli was the first to suggest that probability can be measured fromobservation

P|y/N − p| < ε > 1− δ

Now it seems that there could be a more mathematical treatment ofprobability..

Kolmogorov’s axioms

The axioms and definitions below relate a set Ω called the samplespace and the set of subsets of Ω, F . Every element in E ∈ F iscalled an event

1 If E ,F ∈ F then E ∪ F ,E ∩ F ,E \ F ∈ F . Or more concisely we saythat F is a field of sets.

2 Ω ⊂ F which with the first axiom means that F is an algebra of sets3 Every set E ∈ F is assigned a probability which is a non-negative real

value using the function P : E → [0, 1]4 P[Ω] = 15 If E ∩ F = Φ then P[E ∩ F ] = P[E ] + P[F ], more generally we get

what is called the union bound when E and F are not disjoint thenP[E ∩ F ] ≤ P[E ] + P[F ]

6 If ∩∞n=1En = Φ where En ⊆ En−1 · · · ⊆ E1 we have thatlimn→∞ P[En] = 0. This axiom with axiom 2 allows us to call F aσ-algebra

A random variable x is then understood as a mapping from the size ofelements of F with respect to the probability measure P

Some Set Theory

Kolmogorov’s 6th axiom needs the axiom of choice

Definition (ZFC Axiom of Choice)

It is possible to pick one cookie from each jar even if there is an infinitenumbers of jars

Weird things happen: Can divide a ball into two balls of equal volume

Definition (Axiom of Determinacy)

Every 2 player game with perfect information where two players picknatural numbers at every turn is already determined.

Weird things also happen: We get that there is no such thing asnon-measurable sets

Some Set Theory

Sequential Learning

Von Mises was the first to propose that probability could find itsfoundations in games

Given a bit string 001111

Predict the odds of a 1 (number shouldn’t change much if we look ata subsequence called a collective)

Fortunately we have a method of quantifying how difficult it is topredict the next bit in a string: Kolmogorov complexity!

Sequential Learning

Von Mises was the first to propose that probability could find itsfoundations in games

Given a bit string 001111

Predict the odds of a 1 (number shouldn’t change much if we look ata subsequence called a collective)

Fortunately we have a method of quantifying how difficult it is topredict the next bit in a string: Kolmogorov complexity!

Martingales

Originally Martingales are a gambling strategy that can guarantee a win of1$ given an infinite supply of money

α + 2α + · · ·+ 2iα

stop as soon as you win once

Definition (Martingale)

Given a sequence of outcomes x1, . . . , xn we call a capital process L if

E [L(x1, . . . , xn) | x1, . . . , xn−1] = L(x1, . . . , xn)

L(E )→∞ if E has probability 0 (more on this next slide)Now we define the probability of an event E as

P(E ) = infL0 | limn→∞

Ln ≥ I

Martingales

α + 2α + · · ·+ 2iα

E [L(x1, . . . , xn) | x1, . . . , xn−1] = L(x1, . . . , xn)

Ln ≥ I

Martingales

α + 2α + · · ·+ 2iα

E [L(x1, . . . , xn) | x1, . . . , xn−1] = L(x1, . . . , xn)

Ln ≥ I

Martingales

Theorem (Doob’s inequality)

P[supn

L(x1, . . . , xn) ≥ λ] ≤ 1

Look familiar?

Markov’s inequality!

P[x ≥ λ] ≤ Ex

Other Chernoeff bounds can be derived in this way as well.

Martingales

P[supn

L(x1, . . . , xn) ≥ λ] ≤ 1

Look familiar?

P[x ≥ λ] ≤ Ex

Martingales

P[supn

L(x1, . . . , xn) ≥ λ] ≤ 1

Look familiar?

P[x ≥ λ] ≤ Ex

Bounded Fair Coin Game

We’re now ready to setup the first game between what we the skeptic andnature

Ki is the skeptic’s capital at time i

Mn is the amount of tickets that the skeptic purchases

xn is the value of a ticket (determined by nature)

Theorem

There exists a winning strategy for skeptic but let’s formally define whatwe mean by winning

Winning conditions

We claim that the skeptic wins if Kn > 0∀n and if one two things happen,either

limn→∞

n∑i=1

xi = 0

n→∞Kn =∞

The first condition has two different interpretations in the financelitterature is called a self financing strategy because it means that theskeptic can remain in the game forever without ever having to borrowmoney. Also it means that skeptic wins if nature is forced to playrandomly.The second makes sense, you win if you become infinitely rich butsince this is unlikely this condition embodies the infinitary hypothesiswhich says that there is no strategy that avoids bankruptcy thatguarantees that skeptic become infinitely rich.If P(E ) = 0 then that means that skeptic can become infinitely rich ifE happens

Winning conditions

limn→∞

n∑i=1

xi = 0

n→∞Kn =∞

Winning conditions

limn→∞

n∑i=1

xi = 0

n→∞Kn =∞

The first condition has two different interpretations in the financelitterature is called a self financing strategy because it means that theskeptic can remain in the game forever without ever having to borrowmoney. Also it means that skeptic wins if nature is forced to playrandomly.

The second makes sense, you win if you become infinitely rich butsince this is unlikely this condition embodies the infinitary hypothesiswhich says that there is no strategy that avoids bankruptcy thatguarantees that skeptic become infinitely rich.If P(E ) = 0 then that means that skeptic can become infinitely rich ifE happens

Winning conditions

limn→∞

n∑i=1

xi = 0

n→∞Kn =∞

The first condition has two different interpretations in the financelitterature is called a self financing strategy because it means that theskeptic can remain in the game forever without ever having to borrowmoney. Also it means that skeptic wins if nature is forced to playrandomly.The second makes sense, you win if you become infinitely rich butsince this is unlikely this condition embodies the infinitary hypothesiswhich says that there is no strategy that avoids bankruptcy thatguarantees that skeptic become infinitely rich.

If P(E ) = 0 then that means that skeptic can become infinitely rich ifE happens

Winning conditions

limn→∞

n∑i=1

xi = 0

n→∞Kn =∞

Back to the Fair Coin Game

Theorem

There exists a winning strategy for skeptic

Law of Large Numbers.

Skeptic bets ε on heads, this forces nature not to play heads often or elseskeptic will become infinitely rich. So nature will start playing tails, whenthat happens skeptic puts an ε on tails.

Back to the Fair Coin Game

Theorem

Law of Large Numbers.

Skeptic bets ε on heads, this forces nature not to play heads often or elseskeptic will become infinitely rich. So nature will start playing tails, whenthat happens skeptic puts an ε on tails.

What if xn ∈ [−1, 1] instead of −1, 1?

Theorem

Proof of Bounded Fair Coin Game

We will need some terminology to tackle this problem we define a realvalued function on Ω called P which is a strategy that takes situationss = x1, x2, . . . , xn and decides the number of tickets to buy P(s).

K P(x1x2 . . . xn) = K P(x1x2 . . . xn−1) + P(x1x2 . . . xn)xn

Definition

Skeptic forces an event E if K P(s) =∞∀s ∈ E c

Definition

The skeptic can force

lim supn→∞

n∑i=1

xi ≤ ε

lim supn→∞

n∑i=1

xi ≥ ε

Proof.

take 1 as the starting capital

1 + K P(x1x2 . . . xn) = (1 + K P(x1 . . . xn−1))(1 + εxn) =n∏

(1 + εxi ) < C

where C is a constant so take the log on both sides

n∑i=1

ln(1 + εxi ) ≤ D

Now use ln(1 + t) ≥ t − t2 when t ≥ −1/2

n∑i=1

xi ≤D

εn+ ε

and we get the top part of the lemma. Replace by −ε to get the secondpart.Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 17 / 25

Bounded Forecast games

Somebody has got to be setting the prices, let a forecaster announce priceof ticket at iteration n as mn

Theorem

There exists a winning strategy for skeptic by reduction to the boundedfair coin game.

Proof.

First divide all prices by C to normalize prices to [−1, 1] then set mn = 0and we recover the previous game. Note we also need to change the firstcondition to limn→∞

∑ni=1(xi −mi ) = 0

Bounded Forecast games

Somebody has got to be setting the prices, let a forecaster announce priceof ticket at iteration n as mn

Theorem

There exists a winning strategy for skeptic by reduction to the boundedfair coin game.

Proof.

First divide all prices by C to normalize prices to [−1, 1] then set mn = 0and we recover the previous game. Note we also need to change the firstcondition to limn→∞

∑ni=1(xi −mi ) = 0

Measure theoretic law of large numbers

Assuming Xi are i.i.d random variables with mean µ and variance σ2 wedefine An = X1+X2+···+Xn

n then E [An] = nµn = µ and similarly

Var [An] = nσ2

n2 = σ2/n. By Chebyshev’s inequality we get the weak law oflarge numbers

P( |An − µ| ≥ ε) ≤ Var [An]/ε2 =σ2

To prove Chebyshev’s we define A = w ∈ Ω | |X (w)| ≥ α.X (w) ≥ αIA(w). Take the expectation on both sides to getE (|X |) ≥ αE (IA) = αP(A)In game theoretic proof we don’t need i.i.d assumption we don’t even toassume a distribution exists!

Var [An] = nσ2

P( |An − µ| ≥ ε) ≤ Var [An]/ε2 =σ2

In game theoretic proof we don’t need i.i.d assumption we don’t even toassume a distribution exists!

Var [An] = nσ2

P( |An − µ| ≥ ε) ≤ Var [An]/ε2 =σ2

we don’t even toassume a distribution exists!

Var [An] = nσ2

P( |An − µ| ≥ ε) ≤ Var [An]/ε2 =σ2

To prove Chebyshev’s we define A = w ∈ Ω | |X (w)| ≥ α.X (w) ≥ αIA(w). Take the expectation on both sides to getE (|X |) ≥ αE (IA) = αP(A)In game theoretic proof we don’t need i.i.d assumption we don’t even toassume a distribution exists!

Unbounded game

Theorem

If∑∞

n=1vnn2 <∞ then the skeptic has a winning strategy

Proof.

Similar in nature to proof of the bounded fair coin game. Main idea is thatthe skeptic’s capital is a supermartingale (a sequence that decreases inexpectation)

Unbounded game

Theorem

If∑∞

Proof.

Unbounded game

Theorem

If∑∞

Proof.

What about an application

Suppose you’re a clever young guy/gal who wants to make money off ofthese ideas

A natural next step is to make an infinitely large amount of money off thestock market

Efficient Market Hypothesis

Unfortunately it seems that its difficult to have consistently better returnsthan the market and we will prove this. We make two assumptions thattransaction costs are neglible (not as controversial as it sounds) and thatthe capital of a specific investor isn’t too big relative to the market.

Securities Market Protocol

Proof.

Maybe next time, Finance theory might need its own talk :)

Securities Market Protocol

Proof.

Maybe next time, Finance theory might need its own talk :)

References

Shafer and Vovk (2001)

Probability and Finance It’s only a Game!

Ramon Van Handel

Stochastic Calculus

Peter Clark

All I ever needed to know from Set Theory

Let’s think about how this could change machine learning, talk to me andlet’s write a paper about it!

Probability It's only a game!

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