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Consider the following auction game of complete information (all bidders know the value of other bidders). There are three bidders for single object. Each bidder i ...

Question

Consider the following auction game of complete information (all bidders know the value of other bidders). There are three bidders for single object. Each bidder i has a value Ui for the object. Values are U1 10,U2 5, U3 Each bidder simultaneously and independently submits bid b; € {0,1,2, 1} (bids have to be integers) for the object_Suppose that the winner of the auction has to pay its own bid. For each bidder; identify all weakly dominated strategies.

Consider the following auction game of complete information (all bidders know the value of other bidders). There are three bidders for single object. Each bidder i has a value Ui for the object. Values are U1 10,U2 5, U3 Each bidder simultaneously and independently submits bid b; € {0,1,2, 1} (bids have to be integers) for the object_ Suppose that the winner of the auction has to pay its own bid. For each bidder; identify all weakly dominated strategies.



Answers

Each year an online store can spend at most 50000€ on TV's and laptops. A TV
costs the store owner 500€ and a laptop costs him 750€. Each TV is sold for a
profit of 200€ while laptop is sold for a profit of 350€. The store owner
estimates that at least 15 TV's but no more than 80 are sold each year. He also
estimates that the number of laptops sold is at most half the TV's. How many
TV's and how many laptops should be sold in order to maximize the profit?

It gives us an eBay chart of current bid to minimum bid. Uh, prepare A just says, determine why we know it's a function. And basically, if there's no repeats in the input, it's a function. And so, looking at this, uh, all of the, uh, numerical values for the chart, there's no overlap Each each current bid goes from 49 to $5 in 2014 9 $25 the 99 99 to $100. So because there's no overlap, it is a function. There is no repeats in our inputs for a table. Okay, Now, for part B, it wants us to find, um, find the minimum increments for for $2.50. Um, so I have to do is find the category that holds $2.50 and that category is the first category. So the minimum increment bid is 0.25. So it means that if if the bidding is at $2.50 than the minimum income, it is 25 cents. Okay, Part C just says find at the minimum increment for, ah, 175. That means if the bidding is up to $175. Just like torch are understanding $5 files in that last been. So the minimum bid increase would be $2.50. And then for part D, can we find 400 using this table And because the for, um the table stops at 249 99 we This is undefined. Yeah, on that. So thank you.

Hello suit and went to answer your question about the expected value and using a joint distribution for doing so idea by the problem. In three steps to the first part would be to find the marginal distribution. So you have a table that gives you a probability for education 01 and two 012345. And then what you want to do is for example find the probability if x is zero There will be an in the priority when X is zero on y zero. So in this case then you want to find this case. That would be this value. In this case it will be his value and then you add it. So this would be the column that you want to make here. There is a priority when X is you considering all the cases of way and this is the first value that I found that will be they will go here Then I find the value for x equal to one equal to equal to three equal to four and equal to five. This is known as the marginal distribution because Is the marginal distribution of X. one is equal to zero between all the options of why? Then I do the same process with y. So I said okay let's find the value of when y is zero and x zero when y is zero and X is one and so on. And then I added all the values. So for example for this specific value will be this value and in here there is for one will be this and for the two will be. So in general it always followed this informal. Then I want to find expected value or find an expected value. I want to multiply the specific amount of cars or busses. That is the probability of its successful. For example for the priority of zero cars it's 0.05 and the value of C of course is zero. So it will be sorry I'm starting with way. So the priority of serial busses and the priority this is the relative cirrhosis and the value of serious it will be zero for one bus and the priority associated to one bus. And the probability of two bosses on This is the relative tourists and this is for two. Then I do the same thing for the cars. This will give the expected value for busses and expected value for cars. The final step will be to find the prophet. The prophet is just the price that I pay footage ate them. In your case busses and cars and expected value. So that's the final steps of four busses. You have unexpected but you're still a points decision and the price is still and You have unexpected values. 2.9905 for X or four cars and the price is three and this will give you the total body. So this is a quick reminder when you want to find the profit. You just multiply the price for each good time is expected value For finding the expected value. You multiply the specific value that you are already in use in this case. For example, zero cards. Then the probability that you find with the density function are from finding the density function. You want to find the Well you for this is specific case for zero and will be the density diffusion of zero considering 01 or two. So they fought. The main formula that each situation follows is for the expected value is the cemetery of text. That will be the number of cars and busses times the probability of incredible busses. And for finding the probability you will need to find the probability of this case continued to all the cases of white. So this is a long process, but I hope the explanation was clear and thank you very much for your time.

Let A be the set Consisting of the Numbers 1, 2 3, 4 and five. In task one. We want to determine the truth value of each of the following propositions A. X. Six Eggs. in a search that eggs for three equal 10 for bebe for all X. In a X plus three is less than 10 parsi exists X. In a Such that expert three is less than five. And party for all eggs In a explore three is less than or equal to seven. And we want to find a counter example in B. And see if it exists in task two. We negate each proposition in task one. So let's start with subtask one. Mm So yeah there in proposition A exist X in a With the property that explore 3, 7. The equation X plus three equal seven is the same or is equivalent to X equal Sorry, Explosive three Equal 10. It's equivalent to X equal seven because X if X plus three is equal 10 then it's got to be equal seven. That's the only value that has this property here. The only solution to this equation. So uh the proposition in part they will be true if seven is a. Is an element of the said hey but that's not the case. So proposition is false. Mhm Sim. Political seven is not in the set a. and seven is the only value of X For which explores three equals equals 10. So let's see party for all X. In a explode three is less than 10. We see that explodes three. Less than 10 implies immediately that X is less than 10 -3 equals seven. That is X. Less than seven. So the propositions for all X. M. A. Explore three less than 7 6 equivalent to for all eggs in A. X. is less than seven. This proposition here is equivalent to this one Because the inequality extra three less than 10 In place immediately. That eggs is less than seven. Mhm. In fact we can go from this inequality to this other inequality by um Are in three. Both sides of the inequality. Okay so Let's see if all the elements in a are less than seven. That's true. 1234 and five. All of those numbers Are less than seven. So part B is true. Okay so let's go for a party exist X. In a. Such that X plus three is less than five And we do the same explore three. Less than five. In fact here let's put here equivalent in order to know that they are the same here we use again equivalent. That is because we can pass from one equality to the other by using an algebra operation in this case these three we pass it to the right so X is less than five minus three and that's the same as X. Listen to if we start here we write to as five minutes three and at three both sides and we get this. So they are equivalent. So the proposition for all X in a explosive three, sorry, exists X in a such that X plus three is less than five is the same as exist X in a Such that X is less than two. This proposition is equivalent to this one in part C. And that's because this inequality Is equivalent to this one. And let's see if there is an element in A which is less than two. That is true because we have 11 is less than two and because there is we use the operator exists. We know that it's true. That is at least one element with the property. Then the proposition is true. You're talking about part see here, so part a false party is true and part C is too also. So let's go to party for all X. And a extra three is less than or equal to seven. So we're all eggs in a X plus three Less than a report to seven is equivalent to for all X in a X is less than or equal to the past. These three to the right 7 -3 is four. All the elements. So this will be true if all the elements in a are less than or equal to four but that's not true because there is at least one element which is not less than or equal to four. That is five. So this is false because five is in a N five is not Less than or equal to four. That is his greater. 30 is creator than four as we can see this foot here first. Okay, dad, so in summary for AIDS falls per piece true part is true and party is false. Each time we have a false statement we have a country example that is part A for example exists X in a such as X plus three is 10. In this case we know that that's false because all the elements in a do not verify this equality so we can find country examples in the false cases. In this uh case we want to contract sample in B and B. It's true that is we have no counter examples. That is all the elements in a verify this inequality so there is no kind of example But indeed there is a cut for example. And that counter examples we have found here is five. So let's put it here. B Yes, no, sorry. Yeah, in part B there is no counter example because the proposition there yeah is true and the counter example in for d is five, that is that's the element that contradicts the proposition. There is an element which is not but which has now the property that the element plus three is less than or equal to seven. Okay, so let's go now to subtask number two and we're going to negate all the proposition seen stuck and sub task one. That is in part A I'm going to write only negation. So let's see the proposition for A.S. an element in a with its property experts recall 10, that's the negation of that is all the elements in a failed to have that property. That is all the elements in a Are such, that exploratory is not 10. There is an indication of that's right correctly here, this part negation is just in this case, particular case, the construction has quantified operator exists or for all which is changed to the other in the negation and the investigation of the property in this case, simple thing can be more complex for in this case very simple as that and that's logical because it's if uh proposition is true or false, the other has the of the value of the truth value. So for example, here part A is false and It means that the indication of that is true and the negation, all elements in a has a property that extra Serie is not 10. We can verify it's true because if we had three to all of these elements, we never get the value chain. Okay, as an asian of a nation of B is B is for all X in a extra three at less than 10. So the negation is exists an element in a such that export three is not less than 10. It's not usual to write it this way, but in this other equivalent form that it exists an element in a Such that if express Aries is not less than 10, Then X-plus three is greater than or equal to 10. So this maybe it's a better way to write it, parsi negation of exist on an element in a with the property that expose three is less than five. Negation will be all the elements in A has a property that Explore three East greater than or equal to five. Hicks was three. It's not less than five as I wrote it before. And then I put the equivalent forum which is more is extra three. Sorry, X plus three. It's not less than five. It is created than or equal to five. No party. Yeah, for extra extra sarees less than or equal to 77 delegation is existent X in a such that X-plus three is not less than or equal to seven. That is exists an X in a such debt, Explore three is greater than seven and this proposition is true because this is false and this is true. That is there is an element. We can find an element in a with the property that that element plus three is greater than seven. That element is five because five Plus three is 8 which is greater than seven. Okay, that's it. We have delegation vault. He propositions and we have discussed about the truth value of fish of the given propositions in parts A B C and D. For a fools. Part B is true, part C is true, and parties falls. The counter example in party is five, And here are the negations of each of the propositions in part a 30.

So here in this problem we have to find that how many TVs and the laptop should the store Honor Cell in order to obtain the maximum profit. So we are resuming let X be the number of tv sold in a year. And why be the number of laptop salt in a year? Now we're going to make a constraint. According to the statement, the cost of tv is €500 and the cost of laptop is 7 €50 and the maximum amount amounted, he can spend his €50,000. So we can write that 505 100 X plus 7 50 Y. It should be less than or equals two 50,000. And next statement is that at least 15 TVs but not more than 80 TVs are sold in a year. Silicon say that X should be greater than or equal to 15 and it should be X should be less than or equals to 80. Now. From the next state statement we can conclude that why is less than or equals two X over two. Because it is said that the number of laptops, Soul is at most half the tv number of uh tv. So that is why why is less than or equals two X over two not a maximum profit that we have to obtain us on a single team. We were able to obtain €200 profit and for laptop were able to uh obtain 3 €50. So we can say that the equation for profit would be 200 X. Bliss 3 50 Y. So this is the equation of profit. And these are the constraints. Now we're going to make the graph by using these constraints. So this line represents X is equal to 80 and this is for X is equal to 15 and this line represents 500 X plus 7 50 Y as equals to 15,000. And this line represents why is equal to X over two. No, the Ariel where our solution would lie would be this area and now we'll be obtaining the value of profit by substituting these points. The first corner point is this one and this one this one this one and this one and we obtained that the maximum profit comes out when the corn it is this one that means x it comes out to be as 57 point 143 and why is equal to 28.571 So at this point the profit comes out to be as maximum and its value is 21,428 €0.45. But the number of TVs and laptop it cannot be in fraction. So we can take a number of TVs, we can take it as 57 a number of laptops. You can take a rest 29. So if he sells this amount of this, this this number of TVs and the laptop, he'll able to obtain the maximum profit. So I hope you have understood the problem. Thank you.


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