Applying the Kelly Criterion to Lawsuits Tristan Barnett

Applying the Kelly Criterion to Lawsuits Tristan Barnett Adjunct Lecturer Victoria University

Video Poker Suppose the game is favorable to the player. How much to bet on each trial to maximize the long -term growth of the bankroll? Demonstration Video Poker

Lawsuits Display work agreement The victim was under the impression that he/she was an employee of the company and hence superannuation and holiday pay would apply. When issues were brought up about the type of agreement in 2008, the company stated that he/she was an independent contractor. The company had not issued any tax forms and no tax was taken out.

Video Poker ↔ Lawsuits Is there a connection between the amount to bet in video poker and whether to file a lawsuit over a work agreement dispute? Barnett T, Applying the Kelly Criterion to Lawsuits. Law, Probability & Risk. http: //lpr. oxfordjournals. org/

Analysis of Casino Games A casino game can be defined as follows: There is an initial cost C to play the game. Each trial results in an outcome Oi, where each outcome occurs with profit xi and probability pi. The condition ∑ pi=1 must be satisfied. The percent house margin (%HM) is then -∑Ei/C. The total return is 1+∑Ei/C.

Classical ‘Kelly’ formula Consider a game with two possible outcomes: win or lose. Suppose the player profits k units for every unit wager. Further, suppose that on each trial the win probability p is constant with p+q=1. If kp-q>0, so the game is advantageous to the player, then the Outcome Profit Probability Expected optimal fraction of the current capital to be wagered is given by b=(kp. Profit q)/k. Win $2 0. 35 $0. 70 Lose -$1 0. 65 -$0. 65

‘Kelly’ with Multiple Payouts Theorem 1: Consider a game with m possible discrete finite mixed outcomes. Suppose the profit for outcome i is ki with probability pi for 1 ≤ i ≤ m , where at least one outcome is negative and at least one outcome is positive. Then if a winning strategy exists, and the maximum growth of the bank is attained when the proportion of the bank bet at each turn, b, is the smallest positive root of

‘Kelly’ with Multiple Payouts Proof: Assume a constant proportion b of the bank is bet, with m discrete finite mixed outcomes. Let B(1) / B(0) = 1+ ki b with probability pi for i = 1 to m, where B(t) represents the player’s bank at time t. . Assume the player wishes to maximize g(b) = E[log(B(1) / B(0)) ]. Without loss of generality let k 1 be the maximum possible loss. In the interval 0< b <-1/k 1 1 + ki b > 0 since ki ≥ k 1 for i = 1 to m, so the logarithm of each term is real.

‘Kelly’ with Multiple Payouts (a) g(0) = 0, 0. 1 (b) g′(0) > 0 follows directly from the requirement for a winning strategy, and (c) g′′(b) < 0 for 0<b<-1/k 1 (where k 1 is the MPL) so the first derivate has at most one zero in this interval. Hence whenever there is a winning strategy, the force of growth has a unique maximum given by the root of 0 -0. 1 g(b) -0. 2 -0. 3 -0. 4 -0. 5 b* -1/k 1 b

Video Poker

Video Poker The solver function in Excel is used to calculate this value as b*=0. 030679%. Example: With a $10, 000 bankroll, the Kelly Criterion suggests that the player should initially bet $3. 07 (likely to be round down to $3).

Video Poker The Kelly Criterion can still be applied for a 0. 1 fixed bet by determining the minimum bankroll 0 requirements such that the player is not over betting. If A represents the fixed amount to bet g(b) -0. 1 on each trial and B represents the player’s current bankroll, then a player would not be -0. 2 over betting in the game only if -0. 3 B ≥ A / b* -0. 4 For example, if the “All American Poker” game from Table 3 was fixed at a betting amount of $2. 50 for each trial, then a player would not be over betting in the game if -0. 5 b* -1/k 1 b

“Kelly” revolutions Edward Thorp – Beat the Dealer Edward Thorp – Beat the Market Finding the Edge (2000): Turned over 80 billion on the stock market MIT blackjack syndicate 21 “the movie”

“Kelly” revolutions

Litigation The victim is considering filing a lawsuit against the injurer in an attempt to obtain the total disputed amount of $13, 000. There are risks involved in going to court if unsuccessful. The victim’s chances of recovering the money would likely increase with legal representation. However, there additional legal costs associated with this likely increase in success. It is therefore important to analyze both situations where the victim is representing themselves in court and when a lawyer is acting on the victim’s behalf.

Litigation

Litigation The total legal costs are given by the maximum possible loss (MPL) in the representation of the game. Given the well-defined mathematics of the Kelly criterion, the victim’s decision as to whether to file a lawsuit could be based on B ≥ A / b*. Using Solver in Excel, b*=0. 722. Therefore the victim may consider filing a lawsuit against the injurer if their bankroll is greater than or equal to 1800/0. 722 = $2, 493. In general the victim may consider filing a lawsuit against the injurer if their bankroll is greater than or equal to MPL/b*.

Litigation Using Solver in Excel, b*=0. 647. Therefore the victim may consider filing a lawsuit against the injurer if their bankroll is greater than or equal to 300/0. 647 = $464. It is important to understand the differences in the games given by Tables 5 and 6, and often situations arise where obtaining higher expected costs have more associated risks involved.

Negotiation How much should the victim be willing to accept in an outof-court settlement? Baird D. , Gertner R. and Picker R. (1994), Game Theory and the Law. Expected payout for injurer = $3, 550 Therefore an out-of-court settlement could be $3, 550 + Legal Costs However, the victim has a 30% chance of ending up with a loss in court even though the expected payout is positive. Therefore, the victim may be risk-averse and be willing to accept less than the expected payout in an out-of-court settlement.

Game Theory Minimax Theorem: P 1: A=0. 2, B=0. 8, P 2: A=0. 7, B=0. 3 v = 2. 6 P 1 is guaranteed to win at least 2 by selecting the second row. Von Neumann and Morgenstern (1944) were aware of the relativistic value of the strategies used from the Minimax Theorem. In their words: All this may be summed up by saying that while our good strategies are perfect from the defensive point of view, they will (in general) not get the maximum out of the opponent’s (possible) mistakes, - i. e. they are not calculated for the offensive. It should be remembered, however, that our deductions of 17. 8 are nevertheless cogent; i. e. a theory of the offensive, in this sense, is not possible without essentially

Game Theory The mathematical foundations of game theory as outlined in Von Neumann and Morgenstern (1944) is built around maximizing expectations. This is shown in a game theory axiom which states that numbers in the game matrix must be cardinal utilities and can be transformed by any positive linear function f(x)=ax+b, a>0 without changing the information they convey. f(x) = x+4 P 1: 0. 2 A, 0. 8 B, Possible -3 payout P 1: A=0, B=1, Payout at least 2 P 1: 0. 2 A, 0. 8 B, At least 1 payout P 1: A=0, B=1, Payout at least 6

Game Theory (Kelly Criterion) The Kelly criterion is applied to demonstrate why the favorable player may consider risk-averse strategies. The value of b with both P 1 and P 2 using strategies under the Minimax Theorem is obtained as 0. 222. Therefore P 1 (the favorable player) should wager an amount of 0. 222*current bankroll to maximize the long term growth of the bank. The Kelly Criterion could also be used to determine arbitration based on the size of the bankroll for the favorable player. For example, if P 1 had a bankroll of 4. 38 units, then the risk-averse strategies given below are A=0. 4, B=0. 6. With P 2 playing strategies as A=1, B=0 the arbitrated amount to P 1 is 0. 067. P 1: A=0. 44, B=0. 56 v=0. 11

Game Theory (Prisoner’s Dilemma) If P 1 plays strategy B, then P 1 is guaranteed a positive payout of at least 1 unit regardless of the strategies used by P 2. Whereas playing strategy A could give a player a payout of -1, even though 2 units is possible if both players cooperate. As long as strategy A for P 1 is less than 0. 5, then the expected payout will be positive regardless of the strategies used by P 2. Strategies for P 1 are 0 ≤ A < 0. 5, 0. 5 < B ≤ 1.

Game Theory (Prisoner’s Dilemma) If P 2 plays strategy B, then P 1 will always end up with a negative payout regardless of the strategy P 1 uses. If P 1 plays strategy B, then P 2 will always end up with a negative payout regardless of the strategy P 2 uses. Therefore cooperation is necessary in Game 5, in order for P 1 or both players to obtain a positive payout. Strategies for P 1 and P 2 to obtain a positive payout P 1: 2/3 < A ≤ 1, 0 ≤ B < 1/3 P 2: 2/3 < A ≤ 1, 0 ≤ B < 1/3.

Game Theory (Nash Equilibria) The mixed Nash Equilibria strategies are: P 1: A=1/7, B=6/7 and P 2: A=2/3, B=1/3. The maximin strategies are: P 1: A=4/9, B=5/9 and P 2: A=2/7, B=5/7. Playing the Nash Equilibria mixed strategy of P 1: A=1/7, B=6/7 could result in a negative expected payout if P 2 plays strategy A. For this reason, the maximin strategy could also be identified as a No Regret Policy and by relaxing

Game Theory (Nash Arbitration) It could be argued that P 1 would be more inclined than P 2 to obtain an arbitration process rather than playing the game simultaneously against P 2 to obtain a payout. This is reflected by the observation that P 1 could end up with a negative payout by playing the game simultaneously against P 2, even though the expected payout for P 1 is guaranteed to be positive. Therefore it appears logical that a maximin strategy is used for P 1 to obtain the status quo for the Nash arbitration process. It also appears logical that the mixed Nash Equilibria (assuming one exists) is used for P 2 to obtain the status quo for the Nash

Negotiation The total expected bank = total expected profit x b*. The total expected bank could be used as the minimum amount that the victim should be willing to accept for an out-of-court settlement, and given as $2, 563 from Table 7. Note that this amount is less than the total expected profit of $3, 550.

Arbitration Baird D. , Gertner R. and Picker R. (1994), Game Theory and the Law. Arbitration value = Total Expected Profit + Legal costs = $3, 550 + Legal Costs Barnett T. (2010) Applying Risk Theory to Game Theory, 2 nd Brazilian Workshop of the Game Theory Society, in honor of JOHN NASH, on the occasion of the 60 th anniversary of Nash equilibrium Arbitration value = Total Expected Bank using Kelly Criterion + Legal costs = $2, 563 + Legal Costs

Conclusions The Kelly Criterion (as typically used in favorable casino games) is applied to obtain insights in the decision-making process as to whether it is beneficial for a victim to file a lawsuit against the injurer. The analysis can be used to determine whether a victim should have legal representation or represent themselves in court. Analysis is given to obtain insights as to how much a victim should accept in an out-of-court settlement and a “fair” arbitration value between the two parties in dispute. Casino Mathematics plays a role in solving industry or real-world problems.

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