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Two players, A and B, are invited to take part in a game. The game consists of 10 rounds. In each round, the players will take it in turns to bid against each other for $5, with the first player to bid alternating in each round. The players can bid any amount they choose, subject to the following conditions: the first bid in each round must be at least $1; each subsequent bid must be at least $1 greater than the last bid in that round. Any player can submit a "no bid" at any stage to forfeit the round. At the end of each round, the player with the winning bid pays out the value of their bid and receives the $5.

There are two objectives to the game:

1. Each player should try to maximise their personal winnings

2. Each player should try to win more than their opponent

Player A starts the bidding in round 1.

What is the best strategy for each player to take? Who is the overall winner and how much do they win?

More of a comment than a hint really. Some people may notice that this is similar to a well known game called the dollar auction. However, it is different due to the fact that only the winner pays and due to the two objectives of the game. So, even if you think you know this one, take a look and have a think about it...

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It seems a player would maximize their winnings in each round by making an initial bid of $4.01. Any lower initial bid would simply lead their opponent to win with a lowest legal counteroffer in the range [$4.01 to $5.00]. Subsequent counteroffers would only lose money.

Following this strategy, each player profits $0.99 a round, and the result of the game is a tie, no?

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It seems a player would maximize their winnings in each round by making an initial bid of $4.01. Any lower initial bid would simply lead their opponent to win with a lowest legal counteroffer in the range [$4.01 to $5.00]. Subsequent counteroffers would only lose money.

Following this strategy, each player profits $0.99 a round, and the result of the game is a tie, no?

Um, this problem gives you two objectives. One is to maximize your own profits, the other is to win more than your opponent. Does one of these goals take precedence over the other? Or is there any unambiguous way of telling which of two outcomes is better -- for example, is it better to win $4.50 and do no better than your opponent, or is it better to win $3.10 but beat your opponent by $2?

If winning the most money for yourself is the most important goal, then I agree with mungbean.

If beating your opponent is very very important, then you should bid $4.50. Then either you will gain $0.50 or your opponent will lose $0.50; either way your advantage over your opponent will increase by $0.50. With mungbean's strategy the opponent might bid $5.01 (if losing $0.01 and giving the opponent a $0.01 lead is preferable to gaining nothing and giving the opponent a $0.99 lead).

If gaining money is kind of important, and beating your opponent is kind of important, then the optimal solution might lie somewhere between $4.01 and $4.50.

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I don't see any reason why a rational player would not open each of his rounds with an initial bid of 4.50$ (or 2.50$ or 0.50$). If both players play in this optimal fashion then each player will have won (or lost) the same amount by the end of the game. Depending on each player's choices this could range anywhere from 2.50$ to -2.50$. If a player has the choice of gaining nothing and granting his opponent 0.50$, or paying 0.50$ so his opponent gets nothing, I think he will most likely choose the former since the choice has no impact on the relative game state. Based on this I think it likely that the players will tie with 2.50$ in winnings each.

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It seems a player would maximize their winnings in each round by making an initial bid of $4.01. Any lower initial bid would simply lead their opponent to win with a lowest legal counteroffer in the range [$4.01 to $5.00]. Subsequent counteroffers would only lose money.

Following this strategy, each player profits $0.99 a round, and the result of the game is a tie, no?

Really? Why wouldn't A bid $5.01 in the final round. All he loses is $0.01 against his $4.95 already won, but also ensures he wins the game by denying B his final win.

Also, even assuming A didn't do this, is this really the optimum strategy? You've given a strategy where they tie and end up with $4.95 each - is there no other strategy that also results in a tie but gives each player more? Or that doesn't result in a tie and still gives each player more? Remember the objectives of the game. Your current strategy means that both players fail to achieve either objective.

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Um, this problem gives you two objectives. One is to maximize your own profits, the other is to win more than your opponent. Does one of these goals take precedence over the other? Or is there any unambiguous way of telling which of two outcomes is better -- for example, is it better to win $4.50 and do no better than your opponent, or is it better to win $3.10 but beat your opponent by $2?

If winning the most money for yourself is the most important goal, then I agree with mungbean.

If beating your opponent is very very important, then you should bid $4.50. Then either you will gain $0.50 or your opponent will lose $0.50; either way your advantage over your opponent will increase by $0.50. With mungbean's strategy the opponent might bid $5.01 (if losing $0.01 and giving the opponent a $0.01 lead is preferable to gaining nothing and giving the opponent a $0.99 lead).

If gaining money is kind of important, and beating your opponent is kind of important, then the optimal solution might lie somewhere between $4.01 and $4.50.

You're thinking along the right lines here, but need to carry on that train of thought a bit further. I'd disagree with your answers on the strategy though, for similar reasons pointed out to mungbean.

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I don't see any reason why a rational player would not open each of his rounds with an initial bid of 4.50$ (or 2.50$ or 0.50$). If both players play in this optimal fashion then each player will have won (or lost) the same amount by the end of the game. Depending on each player's choices this could range anywhere from 2.50$ to -2.50$. If a player has the choice of gaining nothing and granting his opponent 0.50$, or paying 0.50$ so his opponent gets nothing, I think he will most likely choose the former since the choice has no impact on the relative game state. Based on this I think it likely that the players will tie with 2.50$ in winnings each.

See my comments to mungbean, as similar principles apply here. I.e. this strategy means that neither player accomplishes either of the objectives.

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See my comments to mungbean, as similar principles apply here. I.e. this strategy means that neither player accomplishes either of the objectives.

In this case I refer to Plasmid's post above. We are given two objectives. Which is more important to a player and by how much?

For example, say my opponent makes an opening bid of 4.25$. I could let him take it, winning him 0.75$ and me nothing, or I could go up to 5.25$ and lose 0.25$ but give him nothing. The former better completes the first objective but the latter better completes the second objective.

And lets say for a moment that the first objective is the most important. Does that mean that it is correct to let my opponent buy with 4.01$ because losing 0.01$ is worse than widening the gap by 0.99$?

Until I figure out what I'm missing I'm gonna stick with always starting every bid at 4.50$ (I take back my earlier statement that starting at 2.50$ or 0.50$ is just as good) with a likely result of each player walking away with 2.50$. I don't think it's possible for a player to accomplish the second objective if his opponent plays properly. The best he can do is come away even. And I don't believe it's possible to make more than 2.50$ if your opponent is competent, greedy and untrustworthy.

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And I don't believe it's possible to make more than 2.50$ if your opponent is competent, greedy and untrustworthy.

So are you saying that a competent, greedy and untrustworthy opponent wouldn't adopt a strategy that would result in them winning more money?

Lets say A opened round 1 with $2.50 as you suggested, but B responds with "no bid". B has just let A win $2.50. B starts round 2 with $2.50. Now A could outbid B with a $4.50 bid (as you've suggested is optimum) and win a further $0.50. However, A can then be pretty certain that B won't let him win anything more than a minimum in the next or subsequent rounds. So by outbidding B, A would be gaining $0.50 but losing at least $2 on the next round. So surely it would make more sense for A (if he is a greedy person) to "no bid" and repay the favour to B, hoping to continue like this throughout the game?

Think about the two objectives more. Is there any way you could apply any weighting to them using only the information given? What if you were playing the game?

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Think about the two objectives more. Is there any way you could apply any weighting to them using only the information given? What if you were playing the game?
Either I'm missing something here, or something's missing.

It seems to me there are 3 ways to approach this:

a) There is a relative value to the two objectives. In which case we need to quantify what that is. Would it be worth losing $1 in order to get the higher score? More information is required in order to determine what the aim of the game is.

b) Maximising winnings takes absolute precedence over beating the opponent. So we would only try to get the higher score if it cost nothing.

c) Beating the opponent takes absolute precedence over maximising winnings. So we would try to get one-up regardless of cost.

If I were playing the game, I would take approach b, since envy has no value to me, but money comes in handy :D

But that's me bringing my own values into the game. By giving people objectives you are defining what their values are within the scope of the game. I'm not inclined to write off one of these objectives as being relatively unimportant, unless you say I can. For example, if you told me the objective was to win as little money as possible, I wouldn't offer an answer which won the most possible, on the basis that I've failed the objective but pocketed lots of cash. Within a game I'll accept the values of the game.

So if you can confirm that the money is of real value to the players whereas beating the other player has infinitessimally small value (no prize or other incentive involved), then we can take approach b and proceed from there. I suspect this is the case but I'd like confirmation before expending too much time going down that avenue.

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If this were a game in real life I was playing I'd be less interested in getting more than my opponent than I would be in making as much money as possible. I might say to my opponent right at the start that I will always let him win the bid for 1$ as long as he lets me win my bids for 1$. Worst case scenario my opponent goes first and is a competitive bastard who doesn't care about the money and he wins 6.50$ while I only win 2.00$ (which is only 0.50$ less than if we had both played 4.50$ openers from the start). Best case scenario I go first and I win 20.50$ and he wins only 16.00$ (because I'm a bastard and I betrayed him when it was too late for him to retaliate).

On the other hand maybe all I care about is getting more than my opponent, in which case if he goes first and offers anything other than 4.50$ for his opening bid I am guaranteed to outscore him. If I go first I'll probably expect to be breaking even because I'll be damned if I give my opponent an opportunity to pull ahead without giving me a chance to catch up like I'd do to him.

If either of these are closer to the correct answer than my original guess, then it seems to me that the question is based entirely on factors not available in the question itself. Namely, my own personal preference on how highly I value each objective and my assessment of how highly my opponent values each objective. If this is the case I don't really see the question even having a correct answer.

Then again maybe it's some kind of sneaky trick question and I'm not seeing the twist...

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I think octopuppy and Tuckleton are getting close but I won't confirm either way just yet and leave you to think about it a bit more, with a couple more hints if you want them...

I believe that you have all you need in the question in order to say what the best strategy is. (You may disagree when we get to the end of this, but hopefully not!)

I mean it when I say "use only the information given in the question". For example, the question does not say that either player has to give back the money at the end.

How many people are involved in the game?

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To me this one seems very similar to the Prisoner's Dilemna. You can always go molehill-for-tat and if both are trying to maximize their take, say offering $4 or $4.01, then after eight rounds both have $4 or $3.96 and player A again bids $4 or $4.01, player B can screw him and bid $1 higher and then bid $4.50 on round 10 to end up with $1 or $.99 more than player A. But player A should see this coming and it circles backward. I think both gaining $2.50 is the most logical way to maximize both objectives with certainty.

EDIT: Attempted to fix censored bit. (how is a molehill atit anyway?)

Edited by plainglazed
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How many people are involved in the game?

Haha! You're a terrible person. I guess the question becomes trivial with 3 (or 9) players. And you were clearly careful in the language you used in the OP (though if you use this again you maybe shouldn't use the word alternate). To be completely honest I was digging in my heels as opposed to trying to keep an open mind about the question. That bit I said about it being a trick question was a joke! I wasn't actually considering it.

Anyways, if there are in fact 3 players and not 2 then the player who goes first should get 2.64$ and the others will get 1.98$.

Hopefully this time it's correct?

Edited by Tuckleton
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Haha! You're a terrible person. I guess the question becomes trivial with 3 (or 9) players. And you were clearly careful in the language you used in the OP (though if you use this again you maybe shouldn't use the word alternate). To be completely honest I was digging in my heels as opposed to trying to keep an open mind about the question. That bit I said about it being a trick question was a joke! I wasn't actually considering it.

Anyways, if there are in fact 3 players and not 2 then the player who goes first should get 2.64$ and the others will get 1.98$.

Hopefully this time it's correct?

It's not a trick question. The question says there are 2 players, A and B. I didn't mean to imply that there may be some other, so far unspoken about players - I'm not that deceptive! However, I was careful with the wording in my "big hint".

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From the hints you've given, I'd say that the second objective is of no importance. You're told it's an objective but you get no prize for it.

Two old enemies keep playing dirty tricks on each other. A says to B "Look, this is silly. Let's resolve all this bickering once and for all, and get all our pent-up frustration out of our systems. We'll have a competition. Let's take it in turns to kick each other in the nuts until one of us can't take it any more. Winner gets $5."

B agrees, and A, since it was his suggestion, gets the first kick. B has to stand, legs wide, eyes closed, ready to take it.

A stands well back, gives it a good run-up, and WHACK! The force lifts B from the ground.

B is doubled up in pain, tears streaming from eyes, face purple.

About half an hour later B manages to get to his feet. "Right..." he winces through gritted teeth, "now it's my turn!"

A smiles and hands over $5. "You win."

So here's a best-case scenario:

Round 1: A starts with "no bid" (B wins $5)

Round 2: B starts with "no bid" (A wins $5)

Round 3: A starts with "no bid" (B wins $5)

Round 4: B starts with "no bid" (A wins $5)

Round 5: A starts with "no bid" (B wins $5)

Round 6: B starts with "no bid" (A wins $5)

Round 7: A starts with "no bid" (B wins $5)

Round 8: B starts with "no bid" (A wins $5)

Round 9: A starts with "no bid" (B wins $5)

Round 10: B starts with "no bid" (A wins $5)

Both win $25.

The last round, as ever, is the problem. But B has very little incentive to bid in the last round, since a bid of less than $4.00 will mean A can ensure B wins nothing.

A bid of over $4.00 could also result in A stopping B from gaining, at some cost to A.

So where human beings are concerned, this strategy is workable.

It would be easier if players can talk freely. Otherwise:

A: Start with "no bid". If on any round the other player won't play along, punish them by raising the bid to $4.50 (or their bid + $1, whichever is greater, even if you lose up to $0.99), then bid $4.50 on your next turn.

B: If other player starts with "no bid", proceed as A. If not, play "no bid" on the 1st round and start the next round with "no bid" in the hope that they catch on (if they do, you might want to play the last round as a copy of A's 1st round. They might let you have that)

By the way, this absolutely wouldn't work for Masters of Logic :D

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OK, Octopuppy's getting close, although there is some more reasoning to justify the answer that hasn't been given yet. Also, if neither player bids then there isn't a "winning" bid. Assume that the first bid having to be a minimum of $1 means that the first player to bid in that round must be at least that much and "no bids" can only come after that. (I won't update the question, as I think the "winning" bid covers this, but I can see where some confusion may come up.)

The last round, as ever, is the problem. But B has very little incentive to bid in the last round, since a bid of less than $4.00 will mean A can ensure B wins nothing.

Is it really just the last round? From what I've said above, if I may assume you'll change your strategy to "$1 bid" - "no bid" each round, then what's to stop B bidding $4.50 in the penultimate round? If he does that, there's no way that A can stop him from winning.

By the way, this absolutely wouldn't work for Masters of Logic :D

Not sure I agree, and that's just made me think of another argument in the other topic, but I'll leave that for another day!

:D
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Is it really just the last round? From what I've said above, if I may assume you'll change your strategy to "$1 bid" - "no bid" each round, then what's to stop B bidding $4.50 in the penultimate round? If he does that, there's no way that A can stop him from winning.
I have completely ignored the second objective. Since there is no incentive to achieve it, why sacrifice potential winnings for it? Hence I have not let it enter into my core strategy at all. The only time it would be of relevance is if the other player fails to see why it is irrelevant, in which case you'll have a pretty poor outcome whatever you do. My strategy may not be perfect in terms of damage limitation in that case, but I've focused on aiming for the best case scenario on the assumption that both players are equally intelligent.

I'm not sure you quite follow my proposed strategy anyway. B will not bid $4.50 in the penultimate round because A starts with "no bid" and B will then get the $5 for nothing. Maybe the rules don't allow that, what do you think?

Admittedly it wouldn't happen in a real auction, so maybe I'm being overambitious.

Assuming that's not allowed, fallback best position:

Round 1: A $1, B "no bid" (A wins $4)

Round 2: B $1, A "no bid" (B wins $4)

Round 3: A $1, B "no bid" (A wins $4)

Round 4: B $1, A "no bid" (B wins $4)

Round 5: A $1, B "no bid" (A wins $4)

Round 6: B $1, A "no bid" (B wins $4)

Round 7: A $1, B "no bid" (A wins $4)

Round 8: B $1, A "no bid" (B wins $4)

Round 9: A $1, B "no bid" (A wins $4)

Round 10: B $1, A "no bid" (B wins $4)

Both win $20.

OK, back to the last round, why wouldn't A bid? Because if s/he did, B would probably prevent her/him from gaining by it. The best "defect" move for A would be to bid $4.50, but this nets her/him $0.50 at most, and in all probability B will bid another $1.00 just to spite her/him. A's potential to grab extra winnings is so little that I doubt s/he would try. It would mean that A achieves the 2nd objective of "winning" the game though surely an informed player would realise how bogus this is.

If you took a more pessimistic view of human nature, you might argue that B should bid $4.50 in that round just to be sure. I don't consider this to be a better move, since it cuts B's winnings to 1/8 of what they could be by sticking to the plan. You'd have to be at least 87.5% certain that A will play the "defect" move in order to justify it.

Similarly if A was extremely pessimistic s/he might think that B would be paranoid and bid $4.50 in the last round. Furthermore s/he may conclude that B, seeing no further basis for cooperation, would "defect" in the 9th round to try to get $0.50 there. If A was at least 87.5% certain that all this would happen, then A would open with $4.50 on the 9th round. But with human players, that's really stretching things.

So I predict that with sensible informed players, the plan is stable. I won't go in detail into what they would do if one player didn't play along at all, it would be a Tіt-for-Tat response along the lines of what I said before. A last round "defection" by A is possible (ie. bid $4.50), and strictly speaking it is a better strategy, but human decency will ensure that it only happens about 87% of the time, which isn't enough to justify a pre-emptive response.

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I can't think of anything new right now that I'm happy with. I look forward to seeing the the answer after the weekend (unless there is still discussion going on.) I'm very curious and more than a little wary of what the eventual solution could turn out to be. Don't you dare disappoint me Neida! :thumbsup:

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I'm not sure you quite follow my proposed strategy anyway. B will not bid $4.50 in the penultimate round because A starts with "no bid" and B will then get the $5 for nothing. Maybe the rules don't allow that, what do you think?

Yeah, I mentioned at the top of my last post that this wouldn't be allowed as there wouldn't be a "winning" bid. You probably didn't notice it as I've put most useful comments in spoilers, but as this was more a clarification I just put it as plain text (as I do now). That's why I assumed you would change your answer to what you've just put. :-)

I can't think of anything new right now that I'm happy with. I look forward to seeing the the answer after the weekend (unless there is still discussion going on.) I'm very curious and more than a little wary of what the eventual solution could turn out to be. Don't you dare disappoint me Neida! :thumbsup:

The pressure!! :-) Octopuppy is pretty much there, although there is some more reasoning to go behind it which I find pretty interesting and confirms the approach. It also leads on to a few interesting points about real life situations... Before I give all the details, I will leave it a little longer and see if anyone gets where I'm going with my "big hint".

Are A and B the only people involved in the game?

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Are A and B the only people involved in the game?
There's also the banker. The banker maybe had a hand in writing the rules, since s/he obviously would want both players to try to be the "winner" :D

And I agree, life is a lot like that. The fashion industry, for instance.

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based on your hints:

is it the case where the no bid player gives the money he bet so far to the banker?

No, this isn't correct. There is nothing in the rules of the game that says this will happen. I have not left out any of the rules of the game.

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OK, it's been a while, so here's my answer for anyone interested...

As I hinted at, there are three people involved in the game. The two players and the "game master" or "bank". I use the term "game master" as the term "bank" suggests only a person dealing with money, whereas someone had to set the rules of the game, manage the game, etc. So "game master" incorporates all those other roles (it could be a person, a company, a machine - whatever).

The first thing for the players to realise is that the two objectives given to them are mutually exclusive. One relies on cooperation, whereas the other relies on competition.

The next thing for them to realise is that they are not really playing against each other, they are each playing against the game master. Think about it - when a player wins a round, who pays out? Not the other player. In fact, they can never win money from the other player. As the only thing they can win from this game is money, then they need to play against the person with the money - i.e. the game master.

Putting the last two points together, if you can't achieve both objectives and so can only aim for one, then aiming to maximise your winnings is the one to go for. Octopuppy jumped straight to this conclusion by saying that was more important than the pyrrhic victory of saying you won (pyrrhic as you have to lose money in order to achieve this). However, when you realise that you are actually playing against the game master, you can see that this objective is solely there for them to create competition between the two players and so there is no reason to aim for it.

So, once all this is realised by both players, they have both seen through the game master and know that cooperation will give the best outcome. So they do what Octopuppy said: $1/no bid, $1/no bid, $1/no bid, through to the end.

Note that the game does rely on trust between the players - any sign this will be broken and they only hurt each other. Octopuppy gave some detail as to what this would mean if they defected on the last round - i.e. the defector would be unlikely to win more in any case.

By the way, I played this game in real life and have seen it played by others. The only additional rule was that each round was limited to 5 bids between the two players. Neither person was allowed to see or talk to the other person - the game master simply took the bids and passed them to the other player. Also, the game master would nod encouragingly whenever a person outbid the other or bid high. He would also frown and give a look like "do you know what you're doing?" when a person bidded low or put in a "no bid". Finally they would announce the result of each round by saying "Well done Player A, you win round 1" (if Player A had the winning bid). The results were fascinating, as the majority of people were prepared to pay over the odds to make sure they "won". In some cases people were bidding thousands to win the round. In every game that I saw the game master ended up winning money (thankfully not real, as otherwise people would have had to get mortgages!) Although a few of us spotted what the best strategy would be, our opponents did not and even by "offering up" rounds to the other player, were never given anything in return, so we simply left the game with nothing. We were told though that, occasionally, both players get it and play cooperatively through to the end, so it can happen!

Final thought - this happens all the time in real life. It's how auctions work for example. OK maybe not auctions for fine art, but auctions for production line stuff definitely. Consider a seller who has 10 items for sale and there are 10 buyers. The seller starts an auction. It would be in each of the buyers interests to just take it in turns to low bid and for the others to let them win. However, trust and competition get in the way - "are you really going to let this go for that? you could sell it for double that yourself!" What if, after the first person has got his, he then continues to bid on the rest knowing that if he gets them for less than market price he can sell them on and make a nice profit? In order to stop that, you can't allow the first person to get such a good deal. So all buyers bid up to about the market price - sometimes more - until a sale is made. Of course, by the end, people who have already bought don't want others to get it any cheaper - they'd rather buy another one cheaper and sell it on if needs be. So everyone ends up paying the greater price.

Another place where this happens is with bankers bonuses: all it takes is for one bank to outbid the others to get all the best staff, so they will all bid high and continue to pay massive bonuses. This is why they are massive in the first place and this is why they're not likely to change - it relies on all the banks changing from their current competitive strategy to a cooperative strategy and it only takes one to break the cooperation and ruin it - there's just not enough trust to go round. Those bankers aren't stupid - they are playing the role of game master! Unfortunately their bosses are!

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