probability it's only a game!
DESCRIPTION
Ever wanted to debate the foundations of probability but got hampered in so many details from set theory to measure theory? This talk explores an alternative foundation to probability theory using game theory instead.The talk starts by giving some historical context for setting the foundations of probability and culminates in an alternative proof of the weak law of large numbers using the bounded fair coin game.TRANSCRIPT
Probability without Measure!
Mark Saroufim
University of California San Diego
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 2 / 25
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$
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 3 / 25
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$
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 3 / 25
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$
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 3 / 25
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..
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 4 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 5 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 6 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 6 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 6 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 6 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 6 / 25
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!
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 7 / 25
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!
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 7 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 8 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 8 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 8 / 25
Martingales
Theorem (Doob’s inequality)
P[supn
L(x1, . . . , xn) ≥ λ] ≤ 1
λ
Look familiar?
Markov’s inequality!
P[x ≥ λ] ≤ Ex
λ
Other Chernoeff bounds can be derived in this way as well.
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 9 / 25
Martingales
Theorem (Doob’s inequality)
P[supn
L(x1, . . . , xn) ≥ λ] ≤ 1
λ
Look familiar?
Markov’s inequality!
P[x ≥ λ] ≤ Ex
λ
Other Chernoeff bounds can be derived in this way as well.
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 9 / 25
Martingales
Theorem (Doob’s inequality)
P[supn
L(x1, . . . , xn) ≥ λ] ≤ 1
λ
Look familiar?
Markov’s inequality!
P[x ≥ λ] ≤ Ex
λ
Other Chernoeff bounds can be derived in this way as well.
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 9 / 25
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)
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 10 / 25
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)
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 10 / 25
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)
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 10 / 25
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)
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 10 / 25
Bounded Fair Coin Game
Theorem
There exists a winning strategy for skeptic but let’s formally define whatwe mean by winning
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 11 / 25
Winning conditions
We claim that the skeptic wins if Kn > 0∀n and if one two things happen,either
limn→∞
1
n
n∑i=1
xi = 0
orlim
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 12 / 25
Winning conditions
We claim that the skeptic wins if Kn > 0∀n and if one two things happen,either
limn→∞
1
n
n∑i=1
xi = 0
orlim
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 12 / 25
Winning conditions
We claim that the skeptic wins if Kn > 0∀n and if one two things happen,either
limn→∞
1
n
n∑i=1
xi = 0
orlim
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 12 / 25
Winning conditions
We claim that the skeptic wins if Kn > 0∀n and if one two things happen,either
limn→∞
1
n
n∑i=1
xi = 0
orlim
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 12 / 25
Winning conditions
We claim that the skeptic wins if Kn > 0∀n and if one two things happen,either
limn→∞
1
n
n∑i=1
xi = 0
orlim
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 12 / 25
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.
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 13 / 25
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.
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 13 / 25
Bounded Fair Coin Game
What if xn ∈ [−1, 1] instead of −1, 1?
Theorem
There exists a winning strategy for skeptic
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 14 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 15 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 15 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 15 / 25
Proof of Bounded Fair Coin Game
Lemma
The skeptic can force
lim supn→∞
1
n
n∑i=1
xi ≤ ε
and
lim supn→∞
1
n
n∑i=1
xi ≥ ε
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 16 / 25
Proof of Bounded Fair Coin Game
Proof.
take 1 as the starting capital
1 + K P(x1x2 . . . xn) = (1 + K P(x1 . . . xn−1))(1 + εxn) =n∏
i=1
(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
1
n
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→∞
1n
∑ni=1(xi −mi ) = 0
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 18 / 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→∞
1n
∑ni=1(xi −mi ) = 0
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 18 / 25
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
nε2
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!
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 19 / 25
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
nε2
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!
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 19 / 25
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
nε2
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!
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 19 / 25
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
nε2
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!
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 19 / 25
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)
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 20 / 25
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)
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 20 / 25
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)
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 20 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 21 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 21 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 21 / 25
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.
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 22 / 25
Securities Market Protocol
Proof.
Maybe next time, Finance theory might need its own talk :)
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 23 / 25
Securities Market Protocol
Proof.
Maybe next time, Finance theory might need its own talk :)
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 23 / 25
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
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 24 / 25
Let’s think about how this could change machine learning, talk to me andlet’s write a paper about it!
Mark Saroufim (UCSD) It’s only a Game! February 18, 2014 25 / 25