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Text of the page (random words):
is an outbreak in a region you do not need to cancel the concerts and soccer games there people will do that for themselves almost nobody wants to get sick if you search the news for south korea and coronavirus you will find many articles about this now let s talk about the u s the public venue closures are too many to count now but look up for a minute and think about the future how does a county in rural kansas decide when to close its schools how does san francisco decide when to reopen theirs most of these closures are nominally until march 31 or april 15 what then it s not going away any time soon effective interventions are targeted targeted interventions are based on data we are late starting this race and at the moment we are still running it blind we need widespread testing to inform both personal and policy response the test does not even need to be very accurate to let us gauge trends and focus efforts assuming it is performed widely enough exponential growth swings both ways small interventions early have outsized effects later at this stage testing is not just an important thing it is the most important thing category uncategorized leave a comment prediction markets and the kelly criterion part 5 nemo 2018 07 15 23 21 perhaps the most famous proponent of the kelly criterion is edward thorp he founded the m i t blackjack club published various papers on gambling and investing and became both a professor of mathematics and a billionaire investor the kelly criterion played a key role in most of these he dubbed it fortune s formula thorp has authored various articles about the kelly criterion over the years e g the kelly criterion and the stock market these typically list six properties related to the kelly formula which i will now attempt to paraphrase if your expected compound rate of growth is positive your wealth will approach infty over time if your expected compound rate of growth is negative your wealth will approach zero over time if your expected compound rate of growth is zero your wealth will approach both infty and zero i e make arbitrarily wide swings over time the ratio between the performance of the kelly strategy and that of any other strategy will approach infty over time the expected time to reach any fixed target wealth is shorter for kelly than for any other strategy to maximize your expected rate of growth over many rounds you can simply maximize the expected logarithm of your wealth each round even if the exact probabilities and payoffs change from round to round at least a couple of these results were first established by thorp himself in the 60s to reiterate the context we assume you have some edge in gambling or investing and you are going to make a large sequence of bets investments using that edge compounding your results over time these properties and the kelly formula itself are about your strategy for sizing each bet if you have no edge you should not be making bets in the first place properties 1 2 and 3 say you do not have to use the kelly formula to do very well smaller or even somewhat larger bets will work fine but be careful not to make your bets too large or you are very surely going to do very poorly property 4 is essentially the one i have mentioned already as the number of bets goes up kelly is increasingly likely to outperform any other strategy and that outperformance is likely to grow toward infty over time property 5 says kelly bets are the fastest expected way to reach a betting investment target property 6 says it is valid to apply the kelly criterion to situations more complex than e g my little toy example with the dice other billionaire investors known or strongly suspected of using kelly methods include warren buffett george soros and james simons that is some impressive company on the flip side the most prominent critic of the kelly criterion was probably paul samuelson a nobel prizewinner in economics he wrote about the kelly formula several times the most amusing surely being his nsf funded academic paper consisting entirely of one syllable words he was presumably trying to make it accessible to the less gifted prof samuelson was apparently a bit of an a hole my kind of guy now nobody likes to laugh at economists more than i do and stodgy academics telling colorful billionaires how to invest certainly seems a ripe opportunity but this is really not fair those same academics would also tell a lottery winner he should never have bought a ticket and they would be right the details and not the outcomes or personalities are what matter we will ponder some of those details in the next installment category finance math leave a comment prediction markets and the kelly criterion part 4 nemo 2018 07 13 18 15 i changed my mind i want to stick with my toy example just a little bit longer let s change the game slightly instead of bringing your own bankroll casino nemo gives you 1 with which to play you can play as many rounds as you like compounding your gains from round to round for as many rounds as you can win in a row and the first time you lose you do not lose your wager you get to keep it but then the game is over and you do not get to play any more so i guess the first time you lose is also the last time there is just one catch you have to pay a one time fee to play question how much should you be willing to pay to play this version of the game i will not bore you with the details but the expectation value of this game is actually infty assuming you go all in on every bet as you should so you pay 1 million to play lose on the fourth round and take home 8 nice work this little thought experiment is called the st petersburg paradox every article about the kelly criterion seems to mention it although they really have very little to do with each other in my opinion but who am i to argue with tradition the first satisfactory solution was provided by bernoulli in 1728 who made the fascinating observation that 100 to a broke man is worth more than 100 to a millionaire in economist speak the utility of money is not linear using simple expectation value as your goal assumes that utility is linear which gives rise to the paradox for expectation value to make sense as a goal it has to be computed over a measurement of value that is linear such a measurement is called a utility function bernoulli decided that a logarithmic utility function was logical i e that the value of any quantity of money equals its percentage of your net worth so 100 to someone with net worth 1000 has exactly the same utility as 100 000 to someone with net worth 1 million equivalently each digit you can tack on to your net worth has the same utility note that defining utility like this is an assertion without proof and not even really you know true we will revisit this when we talk about kelly skeptics using such a logarithmic utility function the st petersburg paradox vanishes because the expectation value is no longer infinite read the wikipedia article for the gory details returning now to a world where you place bets you might actually lose what is the connection between all this and the kelly criterion in kelly s original paper the goal he chose was to maximize the expected rate of return that is given some betting strategy that you apply for n rounds what was your average percentage return per round the strategy that maximizes the expected value of that per round compound return as n becomes large is the kelly strategy note that this is not only a property of the kelly strategy it is the original definition it turns out since percentage return is basically a logarithm and compounding multiplying results is just adding logarithms that this is equivalent to maximizing your expected utility on each round using a logarithmic utility function in fact the wikipedia page for the kelly criterion derives the kelly formula from this fact without really explaining where it comes from or why kelly pointed out in his paper that maximizing the expected logarithm of your bankroll per bet is a consequence of his goal to maximize the compound rate of return and it has nothing to do with any particular concept of utility 1 but that has not stopped lots of people from confusing the two given this defining property of the kelly criterion it is perhaps not so surprising that several people who are famous for their ability to generate large annualized returns are also notable proponents of the kelly criterion we will meet one of them next time 1 logarithmic utility has various implications in this context for example log 0 infty losing one dollar is ok losing your last dollar is very very very bad consequently the kelly formula will not permit any nonzero chance of losing all of your money the formula only tells you to go all in when p 1 i e it s a sure bet if you are in the habit of making such bets you do not need kelly or anyone else to tell you how to size them category finance math leave a comment prediction markets and the kelly criterion part 3 nemo 2018 07 12 22 39 let me continue with my example from part 2 yes this example is a toy but i believe that studying simple cases can help to understand complex ones to recap we have a game where you place a bet that you will win with probability p frac 2 3 and that pays off 1 1 you have a 1000 bankroll to play this game once per day for two days you may compound roll any win loss from the first day into the second we compared three betting strategies strategy a rock go all in always strategy z paper bet nothing always strategy za scissors bet nothing on the first day and go all in on the second changing terminology slightly let s say that one strategy beats another if it is more likely to leave you with more money in a head to head comparison we saw last time that for this two day game paper beats rock and scissors beats paper and rock beats scissors consider one more strategy strategy k bet frac 1 3 of your current bankroll always this is the kelly bet for this game the math is simple when the payoff is 1 1 the kelly formula reduces to p q for this game p frac 2 3 and thus q 1 p frac 1 3 so kelly says to bet frac 2 3 frac 1 3 frac 1 3 of your bankroll note this hypothetical game has positive expectation that is the payoff is more than sufficient to compensate for your chance of losing if you study any actual casino game and plug its numbers into the kelly formula you will get a negative answer which is kelly s way of telling you to take the other side of the bet you can check for yourself that strategy k beats a and it beats z but it loses to za the latter is easy to see since za leaves you with 2000 six times out of nine while the best kelly can do is win twice leaving you with 1000 frac 16 9 1777 i suppose this makes it dynamite blowing up rock and paper while having scissors cut its fuse and we will pretend i designed the example this way on purpose now wait a minute did we just beat the kelly criterion yes yes we did for the two day version of this game but look at strategy za and tell me how to extend it to three days or 10 days 1000 days 1 million days you will find it becomes harder and harder to develop any strategy to beat kelly s simple always bet frac 1 3 this includes adaptive approaches that change strategy based on your win loss record i want to mention again that in all cases strategy a good old rock still has the highest expectation value for example if you come back every year for 100 years and play the 10 day game with strategy a you will probably win the 1 million once or twice which is enough to outrun kelly s expected 2900 per year you will still go bust the other 98 years of course and if we extend the game to 100 days and you stick with strategy a you have to come back for something like 10 17 years for a decent shot at seeing your astronomical payoff and pulling ahead i believe i have now beaten this example into the ground and i am debating what direction to head tune in next time to find out category finance math leave a comment prediction markets and the kelly criterion part 2 nemo 2018 07 11 22 24 welcome to casino nemo you will like it here we have this game where you place a bet and then we roll a fair six sided die if it lands 1 or 2 we keep your bet if it lands 3 through 6 you get back your bet times two 1 1 payoff as i said you will like it here pretend that you have not read part 1 and consider how much should you bet the answer is it depends suppose you are visiting nemoville nemoton nemostan for a ten day vacation and we only let you play this game once per day suppose further that your spouse gives you a strict allowance of 100 each day for gambling it is fairly clear i think that you should bet the entire 100 every day you will probably win around 2 3 of the time so you expect to finish the week with roughly 200 10 frac 2 3 1333 and no other strategy has a higher expectation value in fact the more days you play the better off you become relative to other strategies both in total wealth and in likelihood by betting your entire 100 allowance every day call this strategy of always betting everything strategy a now suppose when you return the following year your spouse changes the rules and gives you a single 1000 allowance for the entire 10 days and you are allowed to compound i e roll your winnings losses forward from one day to the next if you follow strategy a and bet your entire bankroll every day for 10 days there is a 1 frac 2 3 10 98 3 chance you will lose one of the 10 bets and thus all of your money you do have a chance of winning all 10 bets and 1 024 million but that is only 1 7 if we extend this game to 20 or 30 days your chances of winning every bet become vanishingly small very quickly note that the payoff for strategy a if you do manage to win and compound over many days becomes ludicrously huge so huge that this strategy still has a higher expectation value than any other yet if you play it long enough and probably not even very long you will definitely lose everything so perhaps maximizing expected payoff is not the best goal but then what is maybe we can simplify the problem let s reduce your vacation to just two days you have your 1000 allowance and you get to roll your win loss from day 1 into day 2 four things can happen you win both days 4 chances in 9 you win on the first day but lose on the second 2 chances in 9 you lose on the first day but win on the second also 2 chances in 9 you lose both days 1 chance in 9 now strategy a bet it all both days will leave you with 4000 in case 1 and 0 in the other cases for an expected value of 4000 frac 4 9 1778 and this is the highest expectation of any strategy on the other hand strategy a leaves you with nothing more than half the time so maybe you should try something else define strategy z as bet zero always we could say one strategy is better than another if it is more likely to win head to head like say you are on vacation with the neighbors and your spous...