I have devised a system of bargaining that goes like this:
Person A wants to buy an item from Person B. Person B has a value that he will accept for the item, but Person A doesn't know what that value is. Person A makes an offer. If the offer is greater than or equal to the value Person B has decided on, then the sale is completed at that price. However, if the offer is less than the value Person B has in mind, then Person B refuses the offer, AND increases the amount he will take for the item by 1. The amounts are limited to integers in the range 1-100, so if the value is already 100, then it stays at 100.
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Example: Person B has 60 dollars in mind. Person A offers 30 dollars, is declined and the value goes up to 61 dollars. Person A offers 45 dollars, is declined, value is 62. Person A offers 60 dollars, is declined, value is 63. Person A offers 75 dollars, and is accepted, so he pays 75 dollars.
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I have decided that a simple way to determine offers is to have X, an initial offer, and Y, an increment which is added to the offer at each refusal (but capped at 100, of course.) The example above has X=30, Y=15.
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1) How often does Person A pay 100 dollars when X = Y = 25? Express your answer as a percentage or decimal.
2) What is the _average_ amount paid when X = Y = 25?
3) What is the _average_ amount paid when X = 15, Y = 10?
4) Find a function f(X,Y) such that f(X,Y) is equal to the average amount paid in this bargaining system, given initial offer X and increment Y. (Basically just give me a formula to find the average amount for ANY X,Y combo)
5) Find X and Y that produce minimum average this function can offer. I believe I have found the minimum, but if yours is lower, then I'll be able to calculate that.
6) Find a new system that beats the minimum found in (5). I'm not sure if such a system exists, but if you find one, kudos.
Please PM me your answers. Use Logic Puzzle in the subject line.
I will update with answers when I get enough. I may have prizes, if anyone wants to help out.
Person A wants to buy an item from Person B. Person B has a value that he will accept for the item, but Person A doesn't know what that value is. Person A makes an offer. If the offer is greater than or equal to the value Person B has decided on, then the sale is completed at that price. However, if the offer is less than the value Person B has in mind, then Person B refuses the offer, AND increases the amount he will take for the item by 1. The amounts are limited to integers in the range 1-100, so if the value is already 100, then it stays at 100.
.
Example: Person B has 60 dollars in mind. Person A offers 30 dollars, is declined and the value goes up to 61 dollars. Person A offers 45 dollars, is declined, value is 62. Person A offers 60 dollars, is declined, value is 63. Person A offers 75 dollars, and is accepted, so he pays 75 dollars.
.
I have decided that a simple way to determine offers is to have X, an initial offer, and Y, an increment which is added to the offer at each refusal (but capped at 100, of course.) The example above has X=30, Y=15.
.
1) How often does Person A pay 100 dollars when X = Y = 25? Express your answer as a percentage or decimal.
2) What is the _average_ amount paid when X = Y = 25?
3) What is the _average_ amount paid when X = 15, Y = 10?
4) Find a function f(X,Y) such that f(X,Y) is equal to the average amount paid in this bargaining system, given initial offer X and increment Y. (Basically just give me a formula to find the average amount for ANY X,Y combo)
5) Find X and Y that produce minimum average this function can offer. I believe I have found the minimum, but if yours is lower, then I'll be able to calculate that.
6) Find a new system that beats the minimum found in (5). I'm not sure if such a system exists, but if you find one, kudos.
Please PM me your answers. Use Logic Puzzle in the subject line.
I will update with answers when I get enough. I may have prizes, if anyone wants to help out.
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