Game Theory 101 (#47): Lotteries

In , a game is a possibility circulation over outcomes. occur all the time in games because combined methods randomize the outcomes that take place. Therefore, while many energy representations maintain linear buyings, not all energy representations preserve the relative intensity of the choices.

To even control choices in this way, we first need some guidelines for how weigh one lotto versus another. This is where independence and connection enter into play, which we will cover later on.

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  1. Mate, I really need your help. Could you give me 2 examples of why realism cannot represent our world today and 1 reason of why Marxism is not. Just general ideas will do, I can easily build off on them. Thanks please reply ASAP.

  2. Please answer this game theory question that I made up myself, for me. I
    actually don’t know the right answer, even though I made up the
    question. It hurts my head. Thanks.

    2 People (Person A and Person B) are at a very special auction for a
    very special thing. The item in question is a Marble Chess Set worth
    $25. Inside the Chess set there is a random amount of money between $1
    and $800 cash.

    Both People have 1 Bid, and 1 bid only, and Person A bids first. Person B
    can choose not to bid, or can bid $1 higher than Person A and win the
    Chess Set. Person A is given information that the case’s cash money
    amount that is hidden in the case is exactly $700, making the case worth
    $725 total. Person B is given the information that Person A knows how
    much money is in the case, but doesn’t know how much it is himself.

    What is the optimal starting bid for Person A?
    Should Person B bid?

    1. Intro
      For a time I had given this game up. However, in the end I did manage to crack it for an arbitrary number of rounds 🙂 But only approximations can be given as the exact forms are repeating decimals whose fractions get exponentially larger for each successive number of rounds – for example, at 7 rounds the payoffs as fractions already require hundreds of thousands of digits to write. These exact forms quickly become too long to work with, but if they are approximated to thousands of decimal places the calculations can continue reliably.

    2. Explanation
      The calculations themselves come from the fact that if you know the four possible payoffs (starting from no AAA, AAA for you, AAA for opponent, or AAA for both) for a given number of rounds, it is possible to determine all of the frequencies and payoffs with one more round added before it. Each frequency and each payoff for games beginning with each type of round can be written as functions of the four payoffs for the four shorter games they may transition to. In other words, know the payoffs of the four game types, and you can unravel everything else step by step using algebra. So the first thing was to go through the difficult process of deriving all of these formulas and checking and re-checking them. (Technically there do exist expressions for two or more rounds added, but they are usually pages and pages long and much more impractical.)

      By approximating to many decimal places, therefore, with these formulas it is possible to calculate payoffs and frequencies for each number of rounds (beginning with each type of round) one by one with brute force. I find this to be sufficient up to at least 75 rounds or so – as the frequencies converge to specific values, this “depth” gives extremely accurate approximations for them. The difference between the payoffs from one round to the next also converges to a specific value at higher and higher rounds – it is approximately +0.0745251037047180368569191731318479702, so far as I’ve bothered to calculate.

    3. An Example
      For a large number of rounds:
      No AAA
      Play AA about 0.385136, SA 0.344695, and R+P+S 0.270170

      AAA for you
      Play SA about 0.434793, AA 0.303822, and AAA 0.261385

      AAA for opponent
      Play SA about 0.610647, R/P/S 0.244239, and AA 0.145114

      AAA for both
      Play SA about 0.637141, AAA 0.241656, and AA 0.121203

      This is already accurate for about 10+ rounds. The differences at higher rounds occur at more decimal places.

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