Spy Auction

Ahaaa, on the way home from work I think I worked out the optimum answer to the kudos question when complicated bidding instructions are allowed:

The white spy bids 20,000 for the first item auctioned

If he wins that, he bids 25000 for the next item auctioned, otherwise he bids 15000.

If he won either of the two preceding bids, he bids 25000 for the last item auctioned, otherwise he bids 5000 (the black spy only has $5000 left, so cannot overbid him).

I think this bidding method will let him make $5000 more profit than the black spy, regardless of the order this items are auctioned in, and regardless of the items the black spy chooses to overbid him on. It also does not need the bids to be integers.

My working out on the way home was flawed as I forgot the black spy had $50,000 to spend (I thought he only had $40,000) since the black spy has more to bid for this changes things

We define X to be 150000/7 which is approximately $21428.57

The white spy bids X for the first item auctioned

If he wins that, he bids 25000 for the next item auctioned, otherwise he bids 2X-25000 (approximately 17857.14).

If he won either of the two preceding bids, he bids 25000 for the last item auctioned, otherwise he bids 75000-3X (approximately 10714.29) (the black spy only has 75000-3X left, so cannot overbid him).

I think this bidding method will let him make $25000-X (approximately $3571.43) more profit than the black spy, regardless of the order this items are auctioned in, and regardless of the items the black spy chooses to overbid him on. It also does not need the bids to be integers (round the bids up if integer bids are required).[/b]

The bids go in the series 25000*6/7, 25000*5/7, 25000*3/7 until the white spy wins an item, in which case he bids 25000 for all further items. It is worked out that, as the black spy starts with 25000*14/7, he is down to his last 25000*3/7 if he wins the first two auctions so the white spy wins the last auction in that case.

Seems there are many ways to define the problem. For the simple "place up to 100k on the three items, make more profit / less loss than the black, you have twice the money but he knows your bids" things are simple. 25k on everything kills the contest. Any less and it's in black's interest to put in a higher bid; any more and it isn't. With 50k of funds, black can outbid on two items priced below 25k. White uses 75k of cash to get 75k worth of goods, and the contest is a draw. The logic that iamtiger uses makes the situation a lot more interesting, with contingencies in place about the results of earlier bids.

I was thinking this in light of an old dilemma I had heard ages ago. Someone will toss a bomb at you, and you have three choices. Stay in an open field - if the bomb goes there, you have no chance. At home you have a one in six chance of survival if the bomb targets the house. At a bomb shelder, a two in six chance if the bomb hits there. The bomber knows your strategy of how you decide where you go - so what is the optimal strategy? In this case, ideal would be to have only probabilities on where you'll be so that

p (you're at location A) x p (you survive a hit at location A)

is the same for each location.

The black spy does not know how much White will bid; only the instructions. Thus, adding something complex like that denies black the information advantage, and should give White an edge due to having more cash, and therefore, more options - even if the three lots are auctioned simultaneously.

Hmm, so are you saying that 25000/7 (about $3571) profit more than the black spy can be bettered (in the case of him trying to outdo you). Or are you saying that the bids are all auctioned simultaneously so my strategy of making the bid depend on the previous result won't work.

Originally posted by talzamir
Seems there are many ways to define the problem. For the simple "place up to 100k on the three items, make more profit / less loss than the black, you have twice the money but he knows your bids" things are simple. 25k on everything kills the contest. Any less and it's in black's interest to put in a higher bid; any more and it isn't. With 50k of funds, bla ...[text shortened]... cash, and therefore, more options - even if the three lots are auctioned simultaneously.

If we assume that at the optimum survival probability, the chances of the bomber getting you (wherever he shoots) are equal (because otherwise he will bomb he most probable place), then we have enough simultaneous equations to derive the following probabilities for where you might stand:

Open field = 10/37
Home = 12/37
Bomb shelter = 15/37

Which gives him a 10/37 chance of getting you with his bomb.

I will have to think carefully about how this applies to the bidding problem as it is not clear to me.

Originally posted by iamatiger
If we assume that at the optimum survival probability, the chances of the bomber getting you (wherever he shoots) are equal (because otherwise he will bomb he most probable place), then we have enough simultaneous equations to derive the following probabilities for where you might stand:

Open field = 10/37
Home = 12/37
Bomb shelter = 15/37

Which g ...[text shortened]... have to think carefully about how this applies to the bidding problem as it is not clear to me.

Had a think today, if we are relying on randomising the white spy's bid, that implies that the black spy might still get lucky and win, therefore my suggestion which guarantees a return must be better?

Agreed. The best that the randomness can get is a fine expected value and a chance to win.