1/16 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
2/16 ▶ 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
3/16 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. and so 2 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 M ( σ ∗ 1 ) = M ( σ ) − w M ( σ ) = M ( σ ∗ 0 ) + M ( σ ∗ 1 ) . Conversely given M we can defjne the bet: w M ( σ ∗ i ) = M ( σ ∗ i ) − M ( σ ) .
4/16 Success of strategies lim sup n If you can succeed in this way, then it can be shown that you can: lim Realistic considerations: The standard success measure is the condition: M ( X ↾ n ) = ∞ . n M ( X ↾ n ) = ∞ ▶ infjnitely divisible currency ▶ computational restrictions ▶ time to success ▶ infmation Fix a list of acceptable or feasible strategies.
5/16 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
6/16 Saving strategies 2 . 2 ▶ 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 ) ▶ Marginal savings: M ( σ ∗ 0 )+ M ( σ ∗ 1 ) − M ( σ ) . ▶ Success of a saving strategy is harder than success in betting.
7/16 Minimum wagers and saving x ▶ 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 ▶ Scaled strategy succeeds whenever the original strategy succeeds ▶ Plus it achieves savings.
8/16 Minimum wagers and savings paradox 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. ▶ If minimum wager is imposed, saving can hurt a betting strategy ▶ Scaling does not work: scaled-down bets may not be permissible
9/16 Question – Answer – Example Question: What is the rate at which fmuidity of bets needs to increase in order for betting and saving to be equivalent? Example: ▶ Savings paradox is clearly based on the fmuidity restrictions in the bets ▶ Consider games with a variable shrinking minimum bet in each stage Answer: Minimum bet p s at stage s for any ( p i ) with ∑ i p i < ∞ . ▶ At rate 1 / s 2 successful saving is equivalent to successful betting ▶ At rate 1 / s some strategies which don’t have equivalent savings strategy.
10/16 Impossibility of saving with bounded or gauged families of saving strategies. there exists X such that i 2 − g ( i ) = ∞ there exists g-gauged M such that for any Given g with ∑ countable family ( T i ) with one of the following properties: (a) ( T i ) are g-granular saving strategies with bounded total initial capital, i.e. ∑ i T i ( λ ) < ∞ (b) ( T i ) are g-gauged saving strategies ▶ lim sup n M ( X ↾ n ) = ∞ and ▶ lim sup n S T i ( X ↾ n ) < ∞ for each i.
11/16 Countable savings cover when large bets are allowed Given any gauged betting strategy M there exists a countable family the corresponding wagers of M; successfully along X. T i , i ∈ N of saving strategies such that: (a) ( T i ) is computable from M with wagers integer multiples of (b) for each X where lim sup n M ( X ↾ n ) = ∞ there exists i ∈ N such that T i saves successfully along X. Hence for any X where M is successful, at least one of the T i saves
12/16 1 Task: Given savings strategy T, defjne outcome sequence X such that M succeeds and T does not succeed. Proof of impossibility of single savings cover 0 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 t ′ = q ′ · m ′ + r ′ Monitor the division: t = q · m + r → Choose the outcome x so that either q ′ < q or q ′ = q and r ′ ≤ r. if w ≤ p · q x = if w > p · q
13/16 Analysis of outcomes Falure of T along X: Success of M 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 ▶ if outcome is 0 then r ′ < r (from then on) ▶ if r ′ < r at stage s then r − r ′ ≥ p s ▶ the total loss of M for all p s with outcome 0 at stage s is fjnite ▶ since ∑ s p s = ∞ we have that M succeeds along X
14/16 Initial capital and minimum bets But countable cover always exists: Given any g-granular strategy M with initial capital m and any If ∑ i g i < ∞ 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. i g i = ∞ then there are g-granular strategies M with initial capital If ∑ m such that no g-granular strategy can be equally successful with M when they have lesser initial capital. positive x < m, there exists a countable family ( T i ) of g-granular strategies with initial capital x, such that for any X such that lim sup s M ( X ↾ s ) = ∞ there exists i such that lim sup s T i ( X ↾ s ) = ∞ .
15/16 References – Literature Granularity of wagers in games and the (im)possibility of savings Games and Economic Behavior 94 (2015) 157–168 How to gamble against all odds Information and Computation 245 (2015) 152–164 Efgective martingales with restricted wagers https://arxiv.org/abs/1810.05372 Int J Game Theory (2014) 43:145–151 A savings paradox for integer-valued gambling strategies Information and Computation 211 (2012) 160–164 How to build a probability-free casino ▶ Adam Chalcraft, Randall Dougherty, Chris Freiling, Jason Teutsch ▶ Jason Teutsch ▶ Ron Peretz ▶ Gilad Bavly and Ron Peretz ▶ George Barmpalias and Nan Fang
16/16 Why did you work on this? Short answer: Long answer: Thanks for listening! ▶ 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 ▶ 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
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