Savings and initial capital in betting games by George Barmpalias - - PowerPoint PPT Presentation

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Savings and initial capital in betting games

by George Barmpalias (Joint work with Fang Nan) Institute of Software, Chinese Academy of Sciences, Beijing 9th International Conference on Computability and Foundations

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▶ You are in a casino ▶ Betting your money on binary outcomes ▶ Wager is the amount you bet each round ▶ If you win, you double your wager ▶ If you lose, the casino takes your wager

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Modelling betting strategies

A stage in a game is determined by the series of previous outcomes. A strategy is a function that maps any fjnite sequences of outcomes to the amount the player bets at that particular stage. If M(σ) is the capital at σ and we bet w on 0, then w ≤ M(σ) and M(σ ∗ 0) = M(σ) + w and M(σ ∗ 1) = M(σ) − w so

M(σ) = M(σ ∗ 0) + M(σ ∗ 1) 2 . Conversely given M we can defjne the bet: wM(σ ∗ i) = M(σ ∗ i) − M(σ).

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Success of strategies

The standard success measure is the condition: lim sup

n

M(X ↾n) = ∞. If you can succeed in this way, then it can be shown that you can: lim

n M(X ↾n) = ∞

Realistic considerations:

▶ infjnitely divisible currency ▶ time to success ▶ computational restrictions ▶ infmation

Fix a list of acceptable or feasible strategies.

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Efgects of minimum wagers

Suppose minimum bet is $1. ▶ An unlucky row of outcomes may cause bankruptcy ▶ Player is forced out of the game at some fjnite stage ▶ If no minimum bet is enforced, player can play indefjnitely ▶ If losing most of the time, he gradually plays tiny amounts

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Saving strategies

▶ A savings strategy is a betting strategy which also saves money ▶ …by gradually and permanently withdrawing it from the casino ▶ A savings strategy is successful if it eventually saves ∞ ▶ Formally, M(σ ∗ 0) ≤ M(σ) + b and M(σ ∗ 1) ≤ M(σ) − b ▶ Or equivalently M(σ) ≤ M(σ∗0)+M(σ∗1)

2

. ▶ Marginal savings: M(σ∗0)+M(σ∗1)

2

− M(σ). ▶ Success of a saving strategy is harder than success in betting.

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Minimum wagers and saving

▶ If no minimum wager is imposed, saving does not hurt a betting strategy ▶ Why? Any strategy that works with capital x can work with capital x/2 ▶ After withdrawing from playable capital, scale down the remaining bets ▶ If strategy works with initial capital x: ▷ after withdrawing y, bets should be scaled by x−y

x

▶ Scaled strategy succeeds whenever the original strategy succeeds ▶ Plus it achieves savings.

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Minimum wagers and savings paradox

▶ If minimum wager is imposed, saving can hurt a betting strategy ▶ Scaling does not work: scaled-down bets may not be permissible Teutsch: When a fjxed minimum bet is imposed, there exist casinos where betting can succeed but no savings strategy can succeed. Savings paradox: A player can win inside the casino, but upon withdrawing a suffj- ciently large amount out of the game, he is forced into bankruptcy.

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Question – Answer – Example

▶ Savings paradox is clearly based on the fmuidity restrictions in the bets ▶ Consider games with a variable shrinking minimum bet in each stage Question: What is the rate at which fmuidity of bets needs to increase in order for betting and saving to be equivalent? Answer: Minimum bet ps at stage s for any (pi) with ∑

i pi < ∞.

Example: ▶ At rate 1/s2 successful saving is equivalent to successful betting ▶ At rate 1/s some strategies which don’t have equivalent savings strategy.

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Impossibility of saving with bounded or gauged families of saving strategies.

Given g with ∑

i 2−g(i) = ∞ there exists g-gauged M such that for any

countable family (Ti) with one of the following properties:

(a) (Ti) are g-granular saving strategies with bounded total initial capital, i.e. ∑

i Ti(λ) < ∞

(b) (Ti) are g-gauged saving strategies

there exists X such that

▶ lim supn M(X ↾n) = ∞ and ▶ lim supn STi(X ↾n) < ∞ for each i.

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Countable savings cover when large bets are allowed

Given any gauged betting strategy M there exists a countable family Ti, i ∈ N of saving strategies such that: (a) (Ti) is computable from M with wagers integer multiples of the corresponding wagers of M; (b) for each X where lim supn M(X ↾n) = ∞ there exists i ∈ N such that Ti saves successfully along X. Hence for any X where M is successful, at least one of the Ti saves successfully along X.

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Proof of impossibility of single savings cover

Task:

Given savings strategy T, defjne outcome sequence X such that M succeeds and T does not succeed.

Consider a stage of the game and the random variables along X: ▶ t, m are the capitals of T, M ▶ p, w are the minimum bet and wager of T on outcome 1 ▶ t′, m′ are the capitals after the bet takes place and outcome is revealed Monitor the division: t = q · m + r → t′ = q′ · m′ + r′ Choose the outcome x so that either q′ < q or q′ = q and r′ ≤ r. x =        1 if w ≤ p · q if w > p · q

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Analysis of outcomes

Falure of T along X: ▶ q will reach a fjnal value ▶ from then on, the remainder r will be non-increasing ▶ from then on, savings of T are taken from r, so T fails Success of M along X: ▶ if outcome is 0 then r′ < r (from then on) ▶ if r′ < r at stage s then r − r′ ≥ ps ▶ the total loss of M for all ps with outcome 0 at stage s is fjnite ▶ since ∑

s ps = ∞ we have that M succeeds along X

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Initial capital and minimum bets

If ∑

i gi < ∞ then for each g-granular strategy M with initial capital

m, and each positive x < m, there exists a g-granular strategy N with initial capital x, which succeeds exactly where M succeeds. If ∑

i gi = ∞ then there are g-granular strategies M with initial capital

m such that no g-granular strategy can be equally successful with M when they have lesser initial capital. But countable cover always exists: Given any g-granular strategy M with initial capital m and any positive x < m, there exists a countable family (Ti) of g-granular strategies with initial capital x, such that for any X such that lim sups M(X ↾s) = ∞ there exists i such that lim sups Ti(X ↾s) = ∞.

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References – Literature

▶ Adam Chalcraft, Randall Dougherty, Chris Freiling, Jason Teutsch How to build a probability-free casino Information and Computation 211 (2012) 160–164 ▶ Jason Teutsch A savings paradox for integer-valued gambling strategies Int J Game Theory (2014) 43:145–151 ▶ Ron Peretz Efgective martingales with restricted wagers Information and Computation 245 (2015) 152–164 ▶ Gilad Bavly and Ron Peretz How to gamble against all odds Games and Economic Behavior 94 (2015) 157–168 ▶ George Barmpalias and Nan Fang Granularity of wagers in games and the (im)possibility of savings https://arxiv.org/abs/1810.05372

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Why did you work on this?

Short answer: ▶ Fang Nan visited in April and was looking on a project to work on ▶ Teutsch’s paper was interesting to me and wanted to understand it better ▶ Topic is interdisciplinary and many of the researchers are game theorists Long answer: ▶ Many problems in CS and Maths can be formulated in terms of games ▶ Hardness of betting on sequences measures entropy and complexity ▶ Restrictions on betting calibrate hardness and predictability measures ▶ Our result is a theorem in Algorithmic Information Theory ▶ Our result is a separation of two randomness notions Thanks for listening!