Modern Art Strategy I: Auction price for a fixed value painting

Modern Art is one of the first few board games I have played. It is almost a pure auction game, and is best played by 4 or 5 players. Gabriel Rocklin has produced a computer version with AI, which plays competently to some extent. In my opinion, this game is best enjoyed as a party game. It is fun to trash talk your friends into buying something worthless and paying more than he would have liked to at the same time. Nevertheless, for a serious player, one cannot help but wonder, is there a definite strategy for this simple game?

I therefore decided to write a series of articles (probably only two) on the strategy to success in Modern Art.

The Fallacy of EV-1

Unlike some other games on BGG forum, the strategy section for Modern Art is pretty limited. The most useful guide I find, somewhat ironically, is the guide accompanied with the program by Gabriel. In the few articles there, the topic is mostly around how much one should pay for an auction. And the pervasive view is that one should pay EV-1. Unfortunately this is one argument which I am not persuaded at all. Let us review the basic argument:

In an auction, suppose the value of the painting is fixed to be EV. The argument is basically that as a contender, you are always better off getting some profit from the auction rather than being left alone. Therefore, if the bid (from a fellow player) is at x<EV, it is always advantageous for you to bid x+1. This way you get the share (EV-x-1) instead of zero, and the rest of the players get (x+1) instead of EV.

Let us for the time being assume every painting has a fixed value. What is the fallacy of the argument above?

Well, if you want to optimize your earning in this particular earning, then yes, the logic is sound. The problem is, to win the game, optimizing the earning in each auction (dividing between yourself and others) might not be the best idea.

Consider the following example: suppose we have three players, and they have a painting worth $3, $6, $9 in hand respectively. Suppose Bob puts his $9 painting in auction. What will be the best strategy for the second player, Alice, who has a $6 painting to sell?

Clearly, if she thinks by dividing between herself and the rest, it is to her best interest to get $1 instead of $0. However, if every painting is sold for EV-1, clearly Bob is going to win. If Alice wants to have a chance, what she should care instead, is therefore not to let Bob get too much profit. She can afford to lose $1, in exchange of a more even split of the $9 between the other two players.

Suppose she gives up auctioning on this $9 painting. As we will discuss later*, the $3 player will then have no interest to offer higher than $6. Therefore at the end of this auction, the profit will be $3, $0, $6.  Now when Alice sells her painting, the $3 player cannot afford not getting it, and will therefore offer $5. The profit will now be $4, $5, $6. For the $3 painting, Alice can afford to pay $2 for it. In the end, it becomes a tie among the three players. Note that this is not a case of cooperative play; there is no conspiracy involved, and all players are maximizing their own profit without biasing toward other players.

What is the virtue of this little scenario? For the players in the middle of the pack in terms of the quality of the painting they hold, bidding to EV-1 on the more expensive paintings is suicide, and only beneficial to the player with the best holding. He can fare a lot better by letting the bottom players get some profit. Of course, if all players start with equal holdings, then the EV-1 criteria will apply.

*In this specific example, it is easy to say that if you give the top player $7 he is going to win instantly so the maximum is $6. However, there is a more general argument as discussed below.

The Stationary Price

Okay, now that we have seen in an example how EV-1 does not work for the middle player, let us move to another direction: let us consider price without competition. Suppose the auctioneer Tom has to decide between buying the painting on his own or selling it to Tim. What will be the price he starts to be willing to sell it?

This question, I believe, is already answered on the forum, but may be obscured by various reasoning. Let me just straight it out. To Tom’s best interest, he would like to maximize the profit difference between himself and the rest of the players. If he sell it at price xEV, his profit is xEV and the rest of the players average to profit (1-x)/(n-1)EV. If he buys it himself his profit is (1-x)EV and the rest zero. Equating the two, we have

x-(1-x)/(n-1)=1-x; [2+1/(n-1)]x=1+1/(n-1); x=n/(2n-1).

Therefore, without competition, in an average sense (suppose in the subsequent rounds different players will auction) the stationary price of the auction would be n/(2n-1)EV. This assumption is a bit unrealistic however, as if there is no competition, it is quite likely that every time it will be Tim who enters the auction. In this case, to maximize the profit difference between them, Tom will only want to sell at 2/3 EV (set n=1). This is why in the previous section I claimed that the player with $3 will pay at most $6 for the $9 painting.

There is yet another consideration. Suppose there is one single player, Kelly, who is simply buying everything, except possibly her own painting. In this case, how much is she willing to pay? For simplicity assume every player holds the same. Suppose she buys the painting at xEV for every painting except her own, and (conservatively) makes 1/3EV (more than other players) for her own painting. Her total earning will be ((n-1)(1-x)+1/3)EV, whereas the other players make xEV. Equating the two so that she is at least not losing, the upper bound of the price Kelly is willing to pay for each painting is

nx=n-2/3; x=1-2/3n.

For example, in a 4-player game without much competition, one should not bid over 5/6EV if his holding is average; otherwise he will result in a loss even if he wins every auction.

The Bottom Line

What do we learn from all the analyses? Regarding the price of a painting of known value, we can roughly distinguish between three situations:

1. When individual holdings are significantly different, players in the middle of the pack may not want to bid high on the auction  offered by the leading player. The winning bid should range from 1/2EV to 2/3EV, depending on the competition. If there are more than one player lagging behind, it is easy for the leading player to win though, as they will probably compete too much and give the leading player too much profit to overcome.

2. When the holdings are close or not clear, competition will drive the price to EV-1. The catch here is that to really be in this situation you need some serious competition. In particular, there needs to be at least 3 players actively bidding. This is because for those active bidders, they still have to make the majority of their profit from selling their own painting. Therefore they need competition even when they are selling the painting; this makes the least number of active bidders to be 3.

3. When the competition is not fierce for some reason (looking forward to the next article!), one probably has no reason to bid beyond (1-2/3n)EV. This number is still substantial though.

Here comes the one-line conclusion:

In a Modern Art game between experienced players, one expects to make the majority of their profit from selling the painting. The auction price for a painting of value EV, depending on the situation, can range from 1/2EV to (EV-1).

to be continued: how does one play in accor and how does risk shake up the picture