Posted on Categories:Game theory , 博弈论, 经济代写

# 经济代写|博弈论代考Game theory代写|ECON3050 Deleting dominated strategies

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## 经济代写|博弈论代考Game theory代写|Deleting dominated strategies

Every time you throw out a bunch of choices at a node while carrying out a backward induction, you are discarding an equivalent bunch of pure strategies. From the point of view of the strategic form of the game you have reached at that stage, any strategy you discard is dominated by a strategy which is exactly the same except that it calls for a best choice to be made at the node in question.

If we exclude the case when two strategies always yield the same payoff, one strategy is dominated by another if it never yields a better payoff, no matter what strategies the other players may use. Thus hawk dominates dove in the Prisoner’s Dilemma (but not in the Stag Hunt Game of Figure 18).

We can therefore mimic backward induction in a game by successively deleting dominated strategies in its strategic form. We can sometimes reduce a strategic form to just one outcome by this method even when not mimicking backward induction. The result will always be a maximin outcome in a two-person, zero-sum game. But what about games in general?
Any Nash equilibrium of a game you get by eliminating dominated strategies from a larger game must also be a Nash equilibrium of the larger game. The reason is that adding a dominated strategy to your options in a game can’t make any of your current best replies into something worse. You may sometimes lose Nash equilibria as you delete dominated strategies (unless all the dominations are strict), but you can never eliminate all Nash equilibria of the original game.

## 经济代写|博弈论代考Game theory代写|Guessing games

If Alice trades on the stock market, she is hoping that the shares she buys will rise in value. Since their future value depends on what other people believe about them, investors like Alice are really investing on the basis of their beliefs about other people’s beliefs. If Bob plans to exploit investors like Alice, he will need to take account of his beliefs about what she believes about what other people believe. If we want to exploit Bob, we will need to ask what we believe about what Bob believes about what Alice believes about what other people believe.

John Maynard Keynes famously used the beauty contests run by newspapers of his time to illustrate how these chains of beliefs about beliefs get longer and longer the more one thinks about the problem. The aim in these contests was to chose the girl chosen by most other people. Game theorists prefer a simpler Guessing Game in which the winners are the players who choose a number that is closest to two-thirds of the average of all the numbers chosen.

If the players are restricted to whole numbers between 1 and 10 inclusive, it is a dominated strategy to choose a number above 7 , because the average can be at most 10, and $\frac{2}{3} \times 10=6 \frac{2}{3}$. You therefore always improve your chances of winning by playing 7 instead of 8,9 , or 10 . But if everybody knows that, nobody will ever play a dominated strategy, then we are in a game in which the players choose a number between 1 and 7 inclusive. The average in this game can be at most 7 , and $\frac{2}{3} \times 7=4 \frac{2}{3}$. So it is a dominated strategy to choose a number above 5 .

It will be obvious where this argument is going. If it is common knowledge that no player will ever use a dominated strategy, then all the players must choose the number 1.

# 博弈论代写

## 经济代写|博娈论代考Game theory代写|Guessing games

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