Question of Byzantine gamblers

In the impression that the Byzantine Empire is known for its endless court plots and trajectories, it should not deserve this notoriety. Here is a question about a "Byzantine gambler".

The problem is described as follows:

You join a group of advisers in the game. One of the advisors will write 0 or 1 on a piece of paper and show it to other advisors, but you won't know what the number is. They all have superb acting, and you can't infer from their words and applications that he is telling the truth or lying. Each inning, you can choose not to bet, you can also choose to bet on any amount of your assets.

If there are 4 advisors now, two of them will keep telling the truth, but you don't know it's two people. Play three games altogether, your initial fund is 100 yuan, then you can make sure how much money to win?

• A total of four advisers, that first inning will only appear in 2 scenarios: 2 vs 2 or 1 vs 3/0 vs 4.
  • If at least 3 people say the same number, then we can rest assured that 100 dollars will be put on, so that after the end of the first inning or get 200 yuan;
  • If it is a 2VS2, then we choose not to charge the money, so after the end of this sentence, we can know which two are telling the truth, so the back of the money according to what they said.
  • Therefore, the guarantee can win 400 yuan.

If there were only 3 advisers now, and only one would tell the truth. Still play a total of three innings, the initial capital of 100 yuan, then this situation can ensure how much money to win?

• The first inning is either 3 vs 0, or 2 vs 1.
  • If it's 3 vs 0 Then all bets are OK, and if it's 2 VS 1, we don't know who is telling the truth, so we can choose to bet on X-dollar:
    • If the first game lost, that the 1 is the truth of the people, so we have the following 2 innings according to what he said to be able to bet, so you can get 4 * (100-x) yuan;
    • If the first inning wins, then the (100+x) yuan, so that 1 people must be a liar, there is no need to consider him.
      • If the two people in the second inning say differently, we choose not to bet, so that after the end of the second inning will know who is a liar, to the third inning can obtain the 100+x yuan;
      • If the second inning is still the same, all bets will be placed, and the third inning will be analysed as in the second inning.
    • In order to ensure maximum benefit, we should make: 100-x = 100+x, then x = 100/3, guaranteed win 266.66 Yuan

Question of Byzantine gamblers