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Subject: Note on bidding tactics rss

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Olivier Clementin
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1. Expected value of a painting

We define EV as the expected end-of round value of a painting.

Example: if a painting is very likely to finish in the top two, EV will be around 25,000. If a painting will only make it to number three at best, EV will be around 3,000. Assuming all paintings are equally likely to do well, EV would be = (30,000 + 20,000+ 10,000) / 5 = 12,000.

We note that EV is very difficult to estimate because of hidden information and moral hazard. EV depends on a number of factors such as number of paintings by the same artist already on the table (and who owns them), number of paintings by that artist still in the deck, priority order of the artist (LiteMetal being best and Krypto worst), relative stituation of other artists etc... However, players should always try to make an estimate of a painting's EV. Start at 12,000. If the painting is likely to do better than average, EV will be between 12,000 and 25,000. If not, EV will be between 0 and 12,000. If you are lost, assume EV=12,000.

2. Seller's bidding strategy

For the following, we assume that EV can be calculated precisely by all players. We also assume that all players are risk-neutral, i.e. that the value of a painting to a player at any given moment is its EV regardless of uncertainties (on risk neutrality see last paragraph)

We define P the selling price of a painting during an auction.
- If the seller buys the painting for himself, he makes EV - P. All other players make 0
- if another player buys the painting, he makes EV - P. The seller makes P. All others make 0.

This has the following consequences:

Consequence 1: the seller should never buy his painting over half its expected value.
Proof: the seller should buy his own painting only if (EV - P) > P, i.e. only if P < EV / 2.

Consequence 2: the seller is guaranteed to make at least EV/2 from all his sales.
Proof: if someone else bids over EV/2, the seller pockets P which is higher than EV/2. If not, the seller buys the paiting for EV/2 and makes EV - EV/2 = EV/2.

Example: if a painting has an EV of 25,000 (guaranteed to be in the top two), the seller should not bid over 12,500. He will make a minimum of 12,500 from this sale. For the same reason, a seller should NEVER bid more than 15,000 for his own paintings in the first round (even if he is certain that the painting will fetch 30,000). For an "average" placed painting, a seller should bid around 6,000 in the first round.

3. Bidding strategy for other players

Define Non-selling bidders (NSBs) as the players not selling the currently auctioned painting.

First notice that the auction price of a painting should be between EV/2 and EV.
Proof: obviously no one should ever buy a painting higher than EV lest he lose money. Conversely, if the current auction price were below EV/2, a player could bid EV/2 and make more money for himself, while reducing everybody else's payouts.

So, at this point, the final price for a painting could be anywhere between EV/2 and EV. The decision is entirely up to the NSBs. In fact, the NSBs are facing a renewed version of the Prisoner's dilemma.

Theorem: The NSBs are facing a Prisoner's Dilemma with an unstable equilibrium price P*=EV/2 favorable to the players, and a stable (Nash) equilibrium price P**=EV favorable to the seller.

Proof: If the NSBs cooperate, they would want to let the seller buy his painting at P=EV/2, thus ensuring that the seller gets the minimum (and all NSBs get zero). Unfortunately for the NSBs (and fortunately for the seller), this situation is an unstable equilibrium. This is because one of the NSBs will be tempted to bid EV/2 + a small amount S, thus increasing the seller's payout by only a small amount S, and his own payout by a whopping (EV/2 - S). This triggers a chain reaction by which another NSB increase the bid by another small amount S1, thus increasing the seller's payout by S1 and decreasing the NSBs' by S1 (but shifting the reward from another NSB to himself). This chain reaction leads to the stable equilibrium price P**=EV at which the payout is maximum for the seller (payout=P**=EV) and minimum for the NSBs (0). P* is a Nash equilibrium.

4. Tactics for all types of auction

Depending on the type of auction, the seller can move the price from an equilibrium point to another. Usually, the Once Around and the In the Hand auctions will lead to the Nash equilibrium EV because the NSBs cannot cooperate. The Crossed Auction can lead to the unstable equilibrium EV/2 if the NSBs are smart, otherwise the auction will go all the way to the Nash equilibrium at EV. Finally, the optimal strategy for a Fixed Price auction is for the seller to propose 2/3 EV (two thirds of EV). The first NSB will accept it.

- In the Hand: the seller should bid EV/2. The cooperative NSB strategy would be to agree to bid under EV/2, but since agreements are not enforceable, each player will try to get the deal for himself by bidding over EV/2, probably a bit under EV to make some profit. Due to mispricing and overpaying, the final price will probably be around EV.

- Once Around: same as above.The cooperative NSB strategy would be to agree to bid under EV/2, but if they do, since agreements are not enforceable, the last NSB to speak will bid just over EV/2 to get the deal for himself. So all NSBs should forget about cooperation and bid slightly under EV to make some profit. Due to mispricing and overpaying, the final price will probably be around EV.

- Fixed price: The seller can fix the price at 2/3 EV without any risk and the first NSBs will accept it.
Proof: if all NSBs pass, the seller will make EV - 2/3 EV = EV/3. If NSB#1 accepts the deal, he will make EV/3, and the seller will make 2 EV/3. Therefore the incremental value of the deal is EV/3 for the seller and for NSB#1, and 0 for the others NSB. This is clearly a good deal for NSB#1.
Note that if the seller demands more than 2/3 EV he can still get someone to buy it,but it's not assured (unless the game is "loose" -- see below). The NSBs should usually refuse such a deal.

- Crossed Auctions: this is the best opportunity for the NSBs to cooperate and let the seller buy his painting at EV/2. Cooperation is possible because it is possible to react to a NSB breaking the agreement. The best thing the seller can do is to bid low to try and provoke a bidding war between greedy NSBs.

5. "Tight" and "Loose" groups

Different groups of players have different ways of playing. In "loose" groups, players tend to always bid close to EV. In "tight" groups, players tend to bid low and the final price will be closer to EV/2 than to EV.

Some very tight groups never bid over EV/2 as a rule. In such a "disciplined" group, the final price will be closer to EV/2 than to EV, even in In the Hand and Once Around auctions. This is a very rational way of playing. Note, however, that if a loose player joins the group he can win consistently by bidding [EV/2 + something] at all auctions except his own.

As an aside, note that sellers make less money in tight groups. This makes the =type Auction less valuable.

6. Summary

For each type of auction: Optimal seller bid; Optimal NSB bid; Likely price
In the Hand: EV/2; EV-; EV
Once Around: EV/2; EV-; EV
Fixed Price: 2/3EV; 2/3EV; 2/3EV
Cross: EV/2; 0; EV/2

Example1 - Starting painting: EV=12,000
In the Hand: 6,000; 11,000; 12,000
Once Around: 6,000; 11,000; 12,000
Fixed Price: 8,000; 8,000; 8,000
Cross: 6,000; 0 ; 6,000

Example2 - Intrinsic value 60,000, Painting expected to end in top 2. EV = 80,000
In the Hand: 40,000; 75,000; 80,000
Once Around: 40,000; 75,000; 80,000
Fixed Price: 57,000; 57,000; 57,000
Cross: 40,000; 0; 40,000

It appears that In the Hand and Once Around are the best auction types for the seller. Fixed price is not very good. Crossed Auction is risky and can be disastrous for the seller if the NSBs cooperate.

7. Risk aversion

The above calculation assumes that all players are risk neutral. Players who are ahead are risk averse: they should underestimate EV a little. Players who are losing should overestimate EV a little.
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Mark McEvoy
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Re:Note on bidding tactics
olivier6 wrote:
5. "Tight" and "Loose" groups

Different groups of players have different ways of playing. In "loose" groups, players tend to always bid close to EV. In "tight" groups, players tend to bid low and the final price will be closer to EV/2 than to EV.

Some very tight groups never bid over EV/2 as a rule. In such a "disciplined" group, the final price will be closer to EV/2 than to EV, even in In the Hand and Once Around auctions. This is a very rational way of playing. Note, however, that if a loose player joins the group he can win consistently by bidding [EV/2 + something] at all auctions except his own.

As an aside, note that sellers make less money in tight groups. This makes the =type Auction less valuable.


Wha-a-a-a?

Never bid over EV/2 as a RULE, or as guideline?

If by *rule*, then the game will consistently be won by the first to speak on the open auctions. They'll say half of EV and nobody else can/will overcall. Kinda kills the entire point of open auctions - to find that sweet spot where Buyer A would rather give auctioneer X a larger profit for the purpose of denying Buyer B any profityt at all.

If by guideline... well, as you say, the first person to break this guideline will win. Which, to me, should nullify the guideline very quickly. And, to me, makes it a very irrational way of playing. It's the prisoner's dilemma - it only works if nobody else takes their opportunity to step up and win. I don't understand why a group would play a game where there was basically an agreement in place that nobody would step up and try to win by playing a winning strategy.
 
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Rene Wiersma
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Re:Note on bidding tactics
thatmarkguy (#39639),

About "3. Bidding strategy for other players". I think your formula is wrong. A better formula is: P = EV - (EV / N), where N is the number of players.

For example, if the expected value of a painting is 12000 and it is a 4 player game, the painting should sell for 9000. This would mean that the buyer gains 3000 on 2 opponents and loses 6000 on 1 opponent (the seller), which means a net gain of exactly zero. Of course, you don't actually win a game this way, so players should try to bid slightly less than the "right" price, probably 8000 in this case.
 
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Olivier Clementin
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Re:Note on bidding tactics
zaiga (#39760),

Which formula are you referring to ? There is no formula in paragraph 3.

I assume that you suggest an optimal bid for Non Selling Bidders equal to EV - (EV/N). I disagree with you. Let's take your example: NSB#1 is bidding 8,000 with EV=12,000. I am NSB#2: I will bid 9,000. By this bid:
- I make an incremental 3,000 expected profit,
- the seller only makes an incremental 1,000
- NSB#1 loses his 4,000 expected profit

Now NSB#3 will bid 10,000. Why ? because by such a bid
- He makes an incremental 2,000 expected profit
- the seller only makes an incremental 1,000
- NSB#2 loses his 3,000 expected profit.

et cetera...

Therefore, EV-(EV/N) is not an equilibrium. The Nash equilibrium tends to reach EV
 
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Olivier Clementin
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Re:Note on bidding tactics
thatmarkguy wrote:

If by guideline... well, as you say, the first person to break this guideline will win. Which, to me, should nullify the guideline very quickly. And, to me, makes it a very irrational way of playing. It's the prisoner's dilemma - it only works if nobody else takes their opportunity to step up and win. I don't understand why a group would play a game where there was basically an agreement in place that nobody would step up and try to win by playing a winning strategy.


Yes it's by guideline - they tend to stop their bids around EV/2. Don't ask me why they play like that ! It's a group of players who are either disciplined enough to choose the optimal prisoner strategy (in the long run) or stupid enough not to realize that pictures can be worth more than EV/2. I can only suppose that if one of them was breaking the guideline he would only start an inefficient bidding contest.
 
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Rene Wiersma
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Re:Note on bidding tactics
olivier6 (#39767),

Therefore, EV-(EV/N) is not an equilibrium. The Nash equilibrium tends to reach EV

You are right. The trick of the game is then to try and correctly guess the EV and bid accordingly (ie. slightly below the EV).
 
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Re:Note on bidding tactics
olivier6 (#39578),

I've only read as far as 2. Seller's bidding strategy, but I disagree with your formula for how much a seller should bid for thier own painting.
(EV - P) > P is fine for calculating the worth to the seller in isolation to the other players. But shouldn't the gain compared to the other players be the yardstick?

That is, buying my own painting puts me (EV-P)-0 ahead of the other players (I get EV-P, they get 0). And if someone else buys it, I get P-(EV-P) ahead of the buyer (I get P, they get EV-P).
So the break point becomes (EV-P)-0 > P-(EV-P) or buy your own painting for 2/3 the EV.

Of course, you might measure it against the average gain of your opponents, rather than just the buyer. ie, (EV-P)-0 > P-(EV-P)/N which will reach EV/2 if you're playing against an infinite number of players.

I've almost certainly missed something since I don't know nothing about Nash equilibriums and the like. I just use basic math. Any enlightenment?
 
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Olivier Clementin
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Re:Note on bidding tactics
djlg (#40586),

it is true that in some cases, the seller may want to sacrifice some gain in order to decrease the payout of his competitors (for example when the seller his leading by a small margin, closely followed by all other players). In this case, he could increase his price to 2/3 EV as you correctly point out. Note that in actual game information is restricted, so you rarely know when to use this tactics.

However, in the long run, the seller is better off if he consistently bids 0.5 EV, because over (say) 10 auctions he will make 5 EV while his opponents will make the average, or (5/3 x EV) each. On the very long run (over several games), the 0.5 EV bid will appear to be clearly superior.

proof:
over 1000 auctions, a seller who stops at 0.5 EV makes 500 EV, while his opponents make 166 EV each, for a net lead of 433 EV.
over 1000 auctions, a seller who bids 0.666 EV makes 333 EV, while his opponents make 0, for a net lead of 333 EV (inferior)

Conclusion: only in very particular cases in the endgame should you bid over 0.5 EV (and then only if you have enough information). Otherwise, the correct long term strategy is to bid 0.5 EV.

Second thoughts on the fixed price auction:

There is a mistake in my calculation (paragraph 4): At a Fixed price auction the "guaranteed" selling price (i.e. the price which NSBs would be irrational to reject) is actually EV/2, not 2/3 EV. The "optimal" selling price could be beween EV/2 and 2/3 EV depending on the situation, whether the group is "tight" or "loose" and on available information.
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Mark McEvoy
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Re:Note on bidding tactics
olivier6 (#40604),

A followup on rationality ("the price which NSBs would be irrational to reject")...

It would always be irrational for any NSB to refuse a fixed price auction of any item that he expects any later NSB will accept. If any other NSB will accept it, the seller will make the Selling Price (SP) and the buyer will make the difference (EV-SP). It's always better for *you* to be that buyer, than one of your opponents; even if the painting is selling at the ludicrously high price of EV-1 (again, assuming you know or expect someone else would take it at that price).

So then it comes down to what would be rational for the *last* NSB... under the vast majority of circumstances, the price the last NSB would take, rationally, is better than EV/2. Unless he has *very* precise dollar estimates in endgame, he'd rather take ((EV/2)-N) profit and give ((EV/2)+N) profit to the auctioneer, than take 0 profit and give (EV/2) profit to the auctioneer, for most small values of N.

And, in those rare instances where the last NSB wouldn't take the deal, the second-last still has to consider the exact same proposition. And the third last. And so forth. Every player at the table would have to be at the exact number of dollars that accepting the offer would leave them one dollar short of the auctioneer, in order for them to universally refuse a painting at ((EV/2)+1). And even in that case, letting the auctioneer profit from it alone (buy from self at ((EV/2) + 1) and make ((EV/2)-1) profit) will just widen the auctioneer's lead on all of them anyways.

I really can't think of a corcumstance where a fixed price at ((EV/2)+1) would not be a guaranteed sale with rational players.
 
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Re:Note on bidding tactics
olivier6 (#39573),

Holy crap, now I really never want to play this game!
 
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Re:Note on bidding tactics
chaddyboy_2000 wrote:
olivier6 (#39573),

Holy crap, now I really never want to play this game!


Don't be fooled by the apparent depth of the analysis. The truth is that there is no "optimal" play in Modern Art, because

1. there is no optimal solution to the prisoner's dilemma
2. information is restricted
3. aside from bidding tactics (analyzed above), there is a lot of strategy in choosing which painting you sell when it's your turn to sell.
4. psychology (knowing whether your opponents are "loose" or "tight") is much more important than mastering game theory
 
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Olivier Clementin
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Re:Note on bidding tactics
thatmarkguy (#40669),

you are right. My "second thoughts on Fixed price auctions" were wrong, sorry. My initial post is correct: the guaranteed price for Fixed price auction is 2/3 EV.

(Assuming the NSB are rational and behave rationally. For instance, if you suspect that the NSB have sworn an oath never to bid over EV/2, sell at EV/2)
 
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Re:Note on bidding tactics
I note that the tendancy is for the group to become loose.

If the group plays tight, someone who plays slightly looser (bidding such that the auctioneer gains more than him. but he gains over the others), he wins.

Say the art piece is worth 10, and I bid 5. Now its +5 for me, +5 for auctioneer.

The next player faces this situation and realizes: Bidding at least 6 is the way to go, because I am then at +4 against others, -2 against auctionner, instead of -5 against two others.

And the next player would want to bid too.

When it comes to the last guy, if the bid is 8, he would be -2 vs someone, -8 vs auctioneer. It pays to bid 9, since he is then +1 vs others, -8 to auctioneer, a gain.


The tight strategy is NOT a nash equilibrium, since one player changning their strategy without opponents changing will gain an advantage on them.

The nash equilibrium is actually for people to bid EV-1 on most of the types of auctions, since then there is no gain for others to overbid them.

Of course, the auctioneer would not pay more than 1/2 EV.
 
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Re:Note on bidding tactics
I note that the tendancy is for the group to become loose.

If the group plays tight, someone who plays slightly looser (bidding such that the auctioneer gains more than him. but he gains over the others), he wins.

Say the art piece is worth 10, and I bid 5. Now its +5 for me, +5 for auctioneer.

The next player faces this situation and realizes: Bidding at least 6 is the way to go, because I am then at +4 against others, -2 against auctionner, instead of -5 against two others.

And the next player would want to bid too.

When it comes to the last guy, if the bid is 8, he would be -2 vs someone, -8 vs auctioneer. It pays to bid 9, since he is then +1 vs others, -8 to auctioneer, a gain.


The tight strategy is NOT a nash equilibrium, since one player changning their strategy without opponents changing will gain an advantage on them.

The nash equilibrium is actually for people to bid EV-1 on most of the types of auctions, since then there is no gain for others to overbid them.

Of course, the auctioneer would not pay more than 1/2 EV.
 
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Re:Note on bidding tactics
Alexfrog wrote:

Say the art piece is worth 10, and I bid 5. Now its +5 for me, +5 for auctioneer.

The next player faces this situation and realizes: Bidding at least 6 is the way to go, because I am then at +4 against others, -2 against auctionner, instead of -5 against two others.

And the next player would want to bid too.

When it comes to the last guy, if the bid is 8, he would be -2 vs someone, -8 vs auctioneer. It pays to bid 9, since he is then +1 vs others, -8 to auctioneer, a gain.

The nash equilibrium is actually for people to bid EV-1 on most of the types of auctions, since then there is no gain for others to overbid them.



I don't think that works, either. You say " It pays to bid 9, since he is then +1 vs others, -8 to auctioneer, a gain"... IMHO, that's only a gain in a game with more than 8 opponents. Everyone gets roughly equal chance to be auctioneer. So on the long haul, any player that repeatedly takes minus-8 to one player and plus-1 to all others will quickly find themselves minus-something to everyone. In a 4 player game, if you do this in succession on the other three players' auctions you're -6 to all.


The equilibrium place is somewhere in between. I don't think it is 2/3 EV, as I've seen claimed by a lot of people. I think it is variable with the number of players in the game.

In a 5 player game, unless you know the auctioneer is your only rival, it is almost always a good idea to pay 3/4 EV for another player's sale. Take this deal for four other-people's auctions (one per other player) leaves you with (1/4 * 4) = 1 EV profit, to each other player's 3/4 EV profit.

In a 3 player game, unless you know the auctioneer is not a contender, it is seldom a good idea to pay 3/4 EV for another player's sale. Take this deal with each of four other-people's auctions (two per other player) leaves you with (1/4 * 4) = 1 EV profit, to each other player's (2* 3/4) = 3/2 EV profit.
 
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Re:Note on bidding tactics
I don't think that works, either. You say " It pays to bid 9, since he is then +1 vs others, -8 to auctioneer, a gain"... IMHO, that's only a gain in a game with more than 8 opponents.

Think about it. That auctioneer was ALREADY going to get 8 cause someone else bid 8. Now what do you do? You are currently at -8 to the acutionner, -2 to the high bidder, = to the others. If you bid 9, you are -8 to the auctioneer ( no change, he gains 1 and you gain 1), and you gain 1 on the other players but 1+2 on that guy who wouldve gotten it.

Thus, you bidding 9 instead of an opponent bidding 8 means that you are even or better gainst all players, a gain.


Thus, the situation of someone in the group bidding 1/2X, or 2/3X or whatever, and the rest passing, is NOT a nash equilibrium. Any one of those other players could have bid this plus one and made a net gain for themself.

The equilibrium point, if all players have the SAME value for EV, is to bid EV-1, since making a bid of EV negates your gain. Even if you bid EV-1, it still might make sense for some opponent to bid EV, but this gives them a loss of 1 to the auctioneer and a gain of 1 relative to you, in addition to what would already have happened. Thus, some of the bids would go for EV but some would not.



What is interesting is when players calculate DIFFERENT values of EV, that is, they have information in their hands that is valuable to the calculation. Alternately, they know that they can effect the value, such that if they get it they increase the value by playing that type, and if not they refuse to play it. So in fact, the optimal bid is to bid (the highest calculation of EV that will be made by one of your opponents) -1 (or not depending on wether it is worth the risk), IF this bid is less than your EV calculation. Of course, you cant know what others will calculate for the EV, except in a few cases, so this is a judgement thing.

 
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Re:Note on bidding tactics
Alexfrog (#42154),

So, near end of first round, an open double auction comes up that will CLEARLY be worth 30 each (so both = 60)... you open bidding with 59? Because to open any less would be to invite the others to up it by one repeately until one of them hits 59, and you'd rather that person be you than one of your opponents?


I'd expect in a group where everyone plays by the above theory, the game is won at the deal. There's no money to be made in buying, so the game is won by the seller of the most valuable paintings.

I suddenly like the game a whole lot less.
 
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Re:Note on bidding tactics
thatmarkguy (#42271),

Except that:

A) if one player plays against the theory it throws it off for everyone

and

B) even if everyone played the theory perfectly, they will have different ideas about what a painting is worth based on the knowledge that only they have concerning their hand.

-MMM
 
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Re:Note on bidding tactics

I'll agree to B. I just don't like to believe that it is the sole deciding factor (other than the luck of the deal) in Modern Art. I'd like to abstract that out (and abstract out the differences in auction types) - if the game were always for paintings of known resale values, all auctions were open auctions, and all players played 'optimally': would then all paintings sell for EV-1?




But if A is true, then it isn't an equilibrium! Either the player throwing it off is doing so to his own benefit (which would refute the theory that always one-upping all the way up to EV-1 is the optimal play), or the player doing it is simply playing poorly (and I thought you were the one who spanked me in the LotR: Confrontation debate because I assumed one player plays stupid or stubborn in a game theory discussion)

I've long held the theory that, if one player plays tightly in a loose group, or one player plays loosely in a tight group, that player will be at advantage. That's why I've had this as my one and only perfect ten rating. I loved the fact that there is no optimal strategy headed in except to guage what your opponents are doing and adjust accordingly.


But the theory presented by Alexfrog seems to contradict the latter. The theory seems to extend to being that the loosest player is always at advantage.
 
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Re:Note on bidding tactics
thatmarkguy wrote:
I've long held the theory that, if one player plays tightly in a loose group, or one player plays loosely in a tight group, that player will be at advantage. That's why I've had this as my one and only perfect ten rating. I loved the fact that there is no optimal strategy headed in except to guage what your opponents are doing and adjust accordingly.


Let's consider a game where the art is worth 10 and A and B play loose (always bidding 9) and C plays tight (always bidding 5). Let's also assume no one buys their own painting.

After A's first sale:
A +9 B +1 C +0
After B's first sale:
A +10 B +10 C +0
After C's first sale (to A):
A +11 B +10 C +9

After two rounds:
A +21 B +21 C +18

Though it's a close game, C will not catch up if this trend continues.

Now consider a three player game where the art is worth 10 and A and B each play tight (always bidding 5) and player C plays loose (always bidding 9). Let's again assume no one buys their own painting.

After A's first sale:
A +9 B +0 C +1
After B's first sale:
A +9 B +9 C +2
After C's first sale (to A)
A +14 B +9 C +7

After two rounds:
A +23 B +23 C +14

If this goes on C's hole only grows deeper.

So it's false that playing loose is ALWAYS preferable (as shown in the second example).

What should always be happening is that players are playing to the position of the auctioneer. In the second example, players A and B should play looser with C while C is in the hole so as to gain an advantage over the other leader. When the gap closes, tighten up. I believe it is the player that can make THESE judgments the best that ultimately wins the game.

-MMM

 
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Jim Campbell
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Re:Note on bidding tactics
Octavian (#42361) wrote:

So it's false that playing loose is ALWAYS preferable (as shown in the second example)


I just finished walking through some similar examples over in a Goa discussion. One thing to consider when comparing loose and tight bidding strategies is that if opponents bid tightly enough, a solo loose bidder can win by simply buying everything, including the items he auctions. I will duplicate your second example with one change in the loose bidder's strategy. Assume expected values of 10 and that each player auctions one item. A is a loose bidder who will pay up to 9 and will buy their own item if it's a better deal than taking the money. B and C are tight bidders who bid up to 5; they would buy their own for 5 if it was a good move, but with A around that's irrelevant. Profits in each auction are expressed as A/B/C:

A auctions, A purchases for 6: 4/0/0
B auctions, A purchases for 9: 1/9/0
C auctions, A purchases for 9: 1/0/9

Totals: 6/9/9

So A still gives up 3 to the other players with my change, but that's only because we've artificially debilitated A. A is surely capable of noticing his opponent's reluctance to spend and can do much better if he is willing to exploit it by reducing his bids to 7:

A auctions, A purchases for 6: 4/0/0
B auctions, A purchases for 7: 3/7/0
C auctions, A purchases for 7: 3/0/7

Totals: 10/7/7

If A's max bid is 8, the outcome is 8/8/8. I'm struck by the fact that, despite not fully exploiting B's and C's reluctance to pay, A is still able to grab a healthy advantage here. This strategy doesn't just work by itself but has a healthy margin for error.

There is an interesting effect if the tight bidders reduce their max bid below 50% of expected value. Instead of merely balancing the profit of his opponents, A's purchase of his own item becomes another profit center (A's max bid is 7, B and C max bid 4):

A auctions, A purchases for 5: 5/0/0
B auctions, A purchases for 7: 3/7/0
C auctions, A purchases for 7: 3/0/7

Totals: 11/7/7

I was using 4-player examples in my study of this same issue in Goa. Those results show an even greater advantage for a solo loose bidder. Assume that A will bid 7 (and buy his own if the incoming bid is too low) and B, C and D will bid 5:

A auctions, A purchases for 6: 4/0/0/0
B auctions, A purchases for 8: 3/7/0/0
C auctions, A purchases for 8: 3/0/7/0
D auctions, A purchases for 8: 3/0/0/7

Totals: 13/7/7/7

So being the only tight bidder is a disadvantage as MMM demonstrated, but being part of a group of tight bidders playing against one loose bidder is also a disadvantage. That's because the loose bidder can start at a value like EV-2 and then drop to 1 or 2 above the highest that his opponents are willing to pay. Furthermore, if the loose bidders won't bid more than 50% of EV that opens up another avenue for A to exploit.

Returning to 3-player examples: What happens if B and C loosen up a bit, just enough to eliminate the 50% problem and narrow A's profit margin? Assume A will pay 8 and will (quite reasonably) accept 6 instead of buying his own for 7; B and C offer 6 and gladly accept 8:

A auctions, B purchases for 6: 6/4/0
B auctions, A purchases for 8: 2/8/0
C auctions, A purchases for 8: 2/0/8

Totals: 10/12/8

If the purchase of A's items is split over time between B and C, then this situation has approximate parity. Note that in 4-player an analagous example looks like:

A auctions, B purchases for 6: 6/4/0/0
B auctions, A purchases for 8: 2/8/0/0
C auctions, A purchases for 8: 2/0/8/0
D auctions, A purchases for 8: 2/0/0/8

Totals: 12/12/8/8

If A's purchases get spread around in this case, there isn't parity; A is gaining on the other players by 2.7 per round. I investigated the effects as each of the "tighter" bidders defects to the "loose" camp here:

http://www.boardgamegeek.com/article/41962

From that example it seems that the last one to start bidding at least 70% of EV loses.

There are many other factors in an actual auction, such as cash supply and differences in expected value from player to player. But I think simple examples like this lead to some tentative conclusions:

1. Bidding 50% of EV or lower invites the auctioneer to make a gain on everyone by purchasing their own item. Doing it consistently seems to steadily work to the advantage of the players who bid more than 50% of EV. This seems like a significant problem with a tight bidding strategy, since the tight bidders are effectively a group of sheep waiting for a wolf to arrive.

2. If one is the only loose bidder it seems critical to find and use a lower bid than EV-1, since against tight bidders the profit margin on the other transactions is not large enough to make up for the amounts paid to each player. "Testing the waters" early in the game with bids of EV-1 seems like a bad idea, since there might be enough loose bidders to prevent that strategy from achieving parity. Note that it's easy to manage risk as one bids below EV-1, because simply decreasing to EV-2 *doubles* the profit margin. If someone else is playing EV-1 on turn 1, then EV-2 gathers that information; if not, then one can consider moving to EV-3.

3. It seems that the more players there are, the less risky it is to bid close to EV. Based on the above examples, I would consider starting at 70% of EV and then reacting to the other bidders in a 3-player game; in a 4-player game it seems safe to start at 80% and work from there.

Thanks for your help and effort, Matthew. Although I'm more interested in applying this to Goa than Modern Art right now, we're finding out a bunch of interesting stuff.

Jim


 
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Craig Artl
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Re:Note on bidding tactics
chaddyboy_2000 (#40674),

I had to smile when I saw this comment! I just purchased Modern Art and my opinion of all this analysis is similar to yours...it's a game for cripes sake! I'm afraid that I will have to resolve myself to the fact that I will not be qualifying for the Modern Art team when it finally becomes an official Olympic event!
 
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Chris Law
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Re:Note on bidding tactics
jimc (#42411),

I think that the analysis that you did is great but it doesn't take into account that you can overpay people who are losing to make sure that you stay ahead of people that are winning.

So one thing to take into account is that if there is a player ahead of you and 1-2 players behind that you always want to buy from the player(s) that are behind and you should be willing to overbid the player ahead of you in almost all circumstances.
 
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frederick w kruger
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Re:Note on bidding tactics
In resolving the situation with 1 loose bidder(A) and 2 tight bidders(B&C) it seems to me the type of auction plays a role early on.

Assume EV=10 throughout.

For example if the first auction by B is open and C immediately bids 5. A should surely bid 6 first even though she might be willing to bid up to 9. B should not bid up to 9 and C has shown no intention to do so. This makes jumping to 9 seem very stupid.

From here on A knows that C is not willing to bid more than EV/2 and should therefore bid EV/2+1 on all auctions by B.

If however the first auction is in the fist it might take a little longer to sink in what the others are doing but it should still happen.


It has also been mentioned that a single tight player in a loose group will do suffer.

Take a group with A&B loose and C tight. Assume these players either know this before hand or manages to figure it out fairly quickly.

On A's auction B can win with EV/2+1 since C won't go higher. On B's auction A wins with EV/2+1 since C won't go higher. On C's auction A and B fight over the auction and one takes it for EV.

So adding this up gives profits of:
A: EV
B: EV
C: EV

Of course these EV's differ but it is fairly even. This deadlock is however broken on any fixed value auction. Here even a tight player should buy for up to 2/3*EV. In naming a higher price than this it should be left to the artist.

This is hardly convincing that playing tight might be worth it. It does however make it look like a possibility.

In reality I think it is more likely that one would play loose against players that are behind and tighter against players that are ahead.

 
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Brandon Einhorn
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One thing to keep in mind is if a player is employing a loose strategy and has paid a lot for a couple of paintings, its in the groups interst to not sell those paintings. Once there is an element of risk, the expected value goes down. I have seen groups sell established paintings, thus ensuring a nice profit for the early buyers.

By employing the loose strategy, you are taking risk for a small premium. Its up to the opponents to make sure you don't always get your money back.
 
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