I am in a beginner game theory course and I am having some trouble when it comes to games that involve more than 3 players. I don't understand how to set up the games in strategic form and I was wondering if someone could help me out.
One of my home work questions is:

Suppose several friends go out to dinner with the understanding that the bill will be divided equally. The problem is that someone might order something expensive, knowing that part of the cost will be paid by others. To analyze the situation, suppose there are n diners and for simplicity they have the same food preferences. The accompanying table states the price of each of three dishes on the menu and how much each person values it. Value is measured by the maximum amount the person would be willing to pay for the meal.

dish value price surplus
pasta $21 $14 $7
salmon $26 $21 $5
steak $29 $30 -$1

a) suppose there are two diners. What will they order (at a Nash equilibrium)?
b)suppose there are four diners. What will they order (at a Nash equilibrium)?

I solved question a) no problem and got (salmon, salmon) but I have absolutely no idea how to do question b). I know this forum has a rule to try it yourself but I honestly don't know where to start.

Let A;B;C;D be sets in some universal set u. Assume that A U B = u = C U D and
A - C = null set. Prove that u = B U D.

i can get that A is a sub set of C so C U B = u but can't seem to get how you get that to C U D = u. i feel it has something to do with subtracting the absolute compliment of C from D but not really sure.

Counter-example: A = 1, 2; B = 3, 4; C = 1, 2, 3; D = 4.

it isn't just saying A U B = C U D, it is saying that A U B = u = C U D, where u is the set of all possible elements. from the book it seems this is a naive set theory problem so basically we are allowed a set of all sets. so basically A U u = u and A intersection U = A. basically A is some set of elements and B is at least all the elements not in B, same with C and D.

disregard my question, the professor emailed out a correction on the problem and it is really easy. for anyone interested it is A intersection C = null not A - C = null

drinking game question:

I pull one card, opponent pulls 2 cards from a deck. Highest card of the 3 wins, loser drinks once. Either of opponents cards can win.

What % of the time am I drinking?

2/3

lmao that was so incred easy once I thought about it for 3 secs. Thx

Quite a broad question.

If you were to pick one scientific breakthrough in physics which had a large impact on other topics in physics research and other engineering areas. Which would be easy to discuss what would you pick?

F = ma

maybe http://en.wikipedia.org/wiki/Maxwell%27s_equation

How do I solve for x in the following equation?

100 * 1.02 ^ x = 1000 * 1.01 ^ x

1. Divide both sides by 100
2. Take the natural logarithm of both sides

xln(1.02) = ln(10)+xln(1.01)

solve for x.

Have a question for the collective math/statistics genii...

Assume I am playing a slot machine with an expected aggragate payout percentage of 95%. Each spin costs me $10, and I must wager a grand total of $300,000. I of course realize that my expected loss is $15,000.

However, what are my chances of losing $20,000, $25,000...etc.? Conversely, what are my chances of actually coming out ahead? Is it possible to figure out these expectations without knowing the exact paytable and odds for each winning combination (in other words, is knowing just the overall EV enough)?

I'd be very interested to know what the odds are for falling within certain win/loss ranges, and more importantly, how this is calculated. Thanks to anyone who contributes.

You cannot estimate these without knowing the paytable.

Consider two concrete examples.

1) A spin costs you $10, 1 time in a million you win 9.5 million dollars, all other times you lose your $10. Then your expected loss over 30,000 spins is $15,000, but the most common result after 30,000 spins is to have lost $300,000. The chances of you losing over $25,000 is the same as the chances of you losing over $1 - both occur if you fail to hit the 'jackpot'. This happens (999999/1000000)^30000 = 97% of the time.

2) A spin costs you $10. Every time you get back $9.50. Then your expected lost is still $15,000, and the most common result after 30,000 spins is to have lost $15,000. It is impossible that you lose over $25,000. This game is extremely boring to play

Obviously those were extreme and trivial examples, but the point is to show that the EV of a spin is not enough to answer your questions. You also need to know the variance of a spin, at which point you can use the central limit theorem as the number of spins is so large. It seems to me that to calculate the variance you would need the exact paytable.

If this relates to bonus-whoring then you should choose the machine with the smallest variance to maximize your utility. The machine with the smallest variance will tend to be one that pays out relatively frequently with relatively small prizes, rather than one which has a large jackpot which is rarely hit.

ok I need some help please on this calc work, trying to evaluate a derivative and I cant use the shortcut f(x)= x+radical(x)

I started by using lim as h ->0 is (f(x+h)-f(x))/h and have gotten it down to (h+radical(x+h)-radical(x))/h but Im stuck now... Whats my next step?

... conjugate ...

do you mean turn it to h+(x+h)^1/2 -(x)^1/2 all over h, because if so I still dont know what to do next, and if not I dont remember what conjugating is

I got it thanks anyway guys, Im sure ill be back

Im stuck on a couple probability questions

Suppose X is a binomial random variable with n=10 and p=2/5. What is the expected valueof 3X-4?

Probability

let X have the pdf

fx(x) = {2(1-x), 0<= x <=1 or 0, elsewhere}

Suppose that Y=g(x)=X^3 find E(Y) two different ways

Lets H be a subgroup of a group G such that g^-1hg is an element of H for all g element of G and all h is an element of H. show that every left coset gH is the same as every right coset Hg.

Use x = hg <-> h' = g^-1hg = g^-1x <-> x = gg^-1x = gh'.

Last cardcharlie got it right, but remember this theorem as it will become very important.

Probably a really easy question, but how do you solve |1 + i | = ?. And converting re^itheta to a + bi and from a + bi to re^itheta?

|a + bi| = sqrt(a^2 + b^2) = r and b/a = tan(theta).