In an old post, I half-jokingly suggested that the rules of scrabble should be changed to allow the values of tiles to be determined endogenously by competitive bidding. PhD students, thankfully, are not known for their sense of humor and two of Northwestern’s best, Mallesh Pai and Ben Handel, took me seriously and drafted a set of rules. Today we played the game for the first time. (Mallesh couldn’t play because he is traveling and Kane Sweeney joined Ben and me.)
Scrabble normally bores me to tears but I must say this was really fun. The game works roughly as follows. At the beginning of the game tiles are turned over in sequence and the players bid on them in a fixed order. The high bidder gets the tile and subtracts his bid from his total score. (We started with a score of 100 and ruled out going negative, but this was never binding. An alternative is to start at zero and allow negative scores.) After all players have 7 tiles the game begins. In each round, each player takes a turn but does not draw any tiles at the end of his turn. At the end of the round, tiles are again turned over in sequence and bidding works just as at the beginning until all players have 7 tiles again, and the next round begins. Apart from this, the rules are essentially the standard scrabble rules.
Since each players’ tiles are public information, we decided to take memory out of the game and have the players keep their tiles face up. It also makes for fun kibbitzing. The complete rules are here. Share and enjoy! Here are some notes from our first experiment:
- The relative (nominal) values of tiles are way out of line of their true value. The way to measure this is to compare the “market” price to the nominal value. If the market price is higher that means that players are willing to give up more points to get the tile than that tile will give them back when played (ignoring tile-multipliers on the board.) That means that the nominal score is too high. For example, blanks have a nominal score of zero. But the market price of a blank in our play was about 20 points. This is because blanks are “team players:” very valuable in terms of helping you build words. So, playing by standard scrabble rules with no bidding, if the value of a blank was to be on equal terms with the value of other tiles, blanks should score negative: you should have to pay to use them. Other tiles whose value is out of line: s (too high, should be negative), u(too low), v(too low.) On the other hand, the rare letters, like X, J, Z, seem to be reasonably scored.
- Defense is much more a part of the game. This is partly because there is more scope for defense by buying tiles to keep them from the opponent, but also in terms of the play because you see the tiles of the opponents.
- It is much easier to build 7/8 letter words and use all your tiles. This should be factored into the bidding.
- There are a few elements of bidding strategy that you learn pretty quickly. They all have to do with comparing the nominal value of the tile up for auction with the option value of losing the current auction and bidding on the next, randomly determined, tile. This strategy becomes especially interesting when your opponent will win his 7th tile, forcing you the next tile(s) but at a price of zero.
- Because the game has much less luck than standard scrabble, differences in ability are amplified. This explains why Ben kicked our asses. But with three players, there is an effect which keeps it close: the bidding tends to favor the player who is behind because the leaders are more willing to allow the trailer to win a key tile than the other leader.
Finally, we have some theoretical questions. First, suppose there is no lower bound to your score, so that you are never constrained from bidding as much as you value for a tile, the initial score is zero, and there are two players playing optimal strategies. Is the expected value of the final score equal to zero? In other words, will all scoring be bid away on average? Second, to what extent do the nominal values of the tiles matter for the play of the game. For example, if all values are multiplied by a constant does this leave the optimal strategy unchanged?
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June 18, 2009 at 6:29 pm
Michael
This sounds like fun, I might try it sometime. I’ve always thought a similar arrangement might be fun in monopoly, since the sticker values of the properties are quite obviously way off the mark of the true value. Of course, in monopoly credit constraints would be essential.
The asymmetry from the sequential nature of scrabble might prevent the equilibrium from always having an expected value of zero. It would create a wedge between the expected scoring value of a tile between the two players.
It might work something like this: a tile is drawn which is worth either 0 or 3 points to p1 with equal probability and is always worth 2 points to p2. Any equilibrium will have p1 win the auction with a bid somewhere between 2 and 3. His expected score is less than -1/2.
If you included an auction for the right to play first or second I think it would be true.
June 18, 2009 at 6:51 pm
David Pinto
I’m surprised that an ‘S’ should be negative. Since an ‘S’ can be used to turn a six letter word into a seven letter would, or turn a single would into two words, I would think it would be a very valuable letter. I’ve scored very high by playing a single ‘S’ on a double word score, creating two plural words, both getting double value.
Also, does the person who has the next turn bid higher for certain tiles, since that person can see how well he or she will score with that tile?
June 18, 2009 at 8:11 pm
jeff
yes you are right, but that is exactly the reason the value should be negative. compare S with U. Both have a nominal value of 1. Which would you pay more to have? Clearly S. By a lot. So the relative reward you should get for using U must be increased in order to make the scores accurately reflect this.
Another way to think about this, is how difficult is it to use U in a word? Much harder than S. So you should be rewarded more for using U in a word than for using S.
June 18, 2009 at 6:59 pm
Michael
Also, I am curious, since the bids are not allowed to go negative, there seems to be a cap on the difference between nominal value and the real value. Did you find that this constraint was often binding when there were at least two players bidding? For example you say that the nominal values of u and v are too low, does that mean that they usually went for zero?
This seems to be important aspect to your second question. Although I wouldn’t expect that multiplying the nominal values would have any effect, adding a constant to all of the values could make a difference. Alternatively, perhaps negative bids could be allowed?
June 20, 2009 at 10:53 am
Ben
That is likely true; in our experience starting with 100 points was very non-binding in the sense that no one ever got close to approaching zero. While it could happen theoretically it is unlikely in the sense that your bids should be tied to the real values of letters; on the first round it is very unlikely bids would reach higher than say, 75, because it is extremely unlikely to have a work with higher than that many points (of course, this is rough, you could use the letters over sequential turns).
As you said, there is no loss in letting scores go negative. At this point we started at 100 with the 0 cap, but this was just to make things ‘accessible.’ In a game where everyone is trying to win there should be no loss in allowing for starting at 0 and allowing negative scores.
June 22, 2009 at 2:39 pm
Jonathan Weinstein
Hi all,
In response to Jeff’s question of whether the expected final score should be zero: no. Let’s first think about the simplest case, where we each have 6 tiles and are bidding on an “S” which would add about 15 points to either of our “raw” scores compared to the option of losing the auction and receiving the next (random) tile for 0. The auction will end with one of us paying 15 for the “S.” We will both score a net of more than 0, namely the amount we would score without the S. If the “S” would contribute a to one of us and b to the other, then in a two-player game the auction value should be (a+b)/2 (either player can win, no reason for efficiency because defense is as important as offense.) Anyway only the “extra” value of the S goes away in the auction, not the residual value of losing the auction. In particular cases you *could* have zero or negative scores if there was too much inefficiency (players who get much less advantage from premium tiles making defensive bids,) but there is certainly no “zero-score theorem.”
To look at it a more general, non-constructive way, which is more of a proof: suppose the zero-score theorem were true. Then I will deviate from the supposed equilibrium by bidding *slightly* less and losing all auctions. I will get crummy tiles, but score well above zero. My opponent’s score will be virtually unchanged because I’m making him bid almost as much as before. (Note: in a two-player game, the willingness-to-pay of the players is always equal even if one gets much more premium from the tile. I first noticed this in Monopoly auctions.)
Jonathan
June 25, 2009 at 7:32 pm
Market-Based Scrabble « 36 Chambers – The Legendary Journeys: Execution to the max!
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June 28, 2009 at 11:38 pm
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August 11, 2009 at 8:58 am
Michael Giberson
The expected value of the game is not zero for two auction design reasons:
Assume a tile of value A to one player and value B to a second player, with A higher than B, and any other players with values less than B. The first player only has to bid B+1 for the tile (or perhaps as little as B-1), not A, and therefor expects to gain at least A-B-1 from the tile. (In a two person game, with full information including perfect command of the Scrabble dictionary, A = B. With 3+ players valuation seems to become more complex.)
Also, as noted, the last tile up for auction each round is non-competitively assigned to the player needing a 7th tile at zero price, but will have a positive value. This expected value then also constrains the bids for the second-to-the-last tile up for auction each round, because winning that tile forecloses the opportunity to get the expected value of the last tile, and so on working backward through the tiles up for auction.
Using a simultaneous ascending auction would “cure” the second problem because there would be no non-competitively assigned tiles. Turn over the appropriate number of tiles and then have players begin bidding. A notebook will be handy.
August 11, 2009 at 11:54 am
Michael Giberson
With a modest amount of reflection, I see that my thinking in the “Assume a tile of value A” logic is off (but for the parenthetical remark at the end, which gets it right). As Jonathan said, in a two player game the willingness-to-pay for a tile is the same for both players. In the simple case of only two tiles left to bid for in a round, all that matters for the valuation is the effect on the net score comparing [player-1 gets current tile, player-2 gets next tile] to [player-2 gets current tile, player-1 gets next]. The net effect on game score is identical for both players.
The valuation logic undermines my “Also, as noted” paragraph argument, since the value of the non-competitively assigned tile becomes embedded in the price paid for the second-to-the-last tile. No free lunches.
Backward induction yields a general ‘no free lunches’ condition, which implies an expected net score of zero (at least if the bid constraints are never binding).
August 23, 2009 at 8:11 am
Hundred Pockets» Blog Archive » Is nothing sacred?
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November 18, 2009 at 2:37 am
Anonymous
You should enter the wonderful world of Eurogames; a significant portion of them are auction-based.
March 20, 2014 at 10:25 pm
Parivesh
Hi guys, I am so disappointed that this app is not avalbaile in the UK android market. May I ask why? My wife and I love a good game of scrabble and would love you to make it avalbaile over the pond! Please tell me why so I can at least understand why this App is not avalbaile here. Thanks and sorry for the whinge!
November 18, 2009 at 11:05 am
unekdoud
I’ve actually seen Monopoly with bidding, however it is quite as Jonathan has stated above: players bid very near to the true value to keep each other out. This also means that players have to start with thousands of dollars to keep them from going into the negative(enforcing this makes the strategies very interesting). On the other hand, Monopoly is quite luck-based, and having a few players with too much money turn the game into an endless random walk (If I’m not wrong, the average gain is positive, so expect to play forever…)
November 19, 2009 at 4:17 am
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November 21, 2009 at 12:15 pm
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[…] Says Jeff Ely on his blog Cheap Talk: At the beginning of the game tiles are turned over in sequence and the players bid on them in a fixed order. The high bidder gets the tile and subtracts his bid from his total score. (We started with a score of 100 and ruled out going negative, but this was never binding. An alternative is to start at zero and allow negative scores.) […]
November 24, 2009 at 9:19 am
Rob S
In Scrabble (played by the normal rules) players replenish their tiles after each play, and thus they have time during their opponents turns to consider available plays. Notwithstanding that some players are faster than others, each player has a similar amount of time to think, and there is little or no “down-time”. This appears to be lost in Auction-Scrabble.
From the perspective of a hobby gamer, the sequence of play looks questionable:
After the auction phase is completed, Player-1 makes a play on the board and then must wait while each other player takes a turn. Only when a new auction commences will Player-1 get to act again. Notwithstanding the possibility of kibitzing other players, this looks dull.
It would be interesting if there was more detail here: “Scrabble normally bores me to tears but I must say this was really fun.” This is the only statement about the experience of playing the game: I would like to know what made it fun, how long the players took to finish, and the circumstances of their game.
As a consequence of the downtime inherent in multi-player games, Auction-Scrabble looks best as a two-player game. This is a pity, as auction mechanisms are typically more entertaining with larger player numbers. Interestingly, Scrabble enthusiasts generally prefer to play with no more than two players – this reduces the amount of unknown information (tiles hidden on opponents racks) and downtime.
As a committed games enthusiast, it is interesting to read so many comments which focus on the economics of bidding in a game, rather than giving primary consideration to the /playability/.
Rob
March 30, 2010 at 6:56 am
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May 4, 2010 at 2:02 pm
john baker
Hey, if you are interested in playing scrabble with friends and family, check out our games site. (http://myemailgames.com/playgamesl.aspx) The game will allow you to play scrabble on the web with up to 4 players. When your turn is finished it will send an email to the next player’s turn. Lots of people are playing it already. If you have any questions email info@myemailgames.com
August 2, 2011 at 2:21 pm
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August 17, 2011 at 8:50 am
Patrick
Regarding the comparison of nominal value and market price:
I wonder if it shouldn’t be the other way round. In my opinion, the nominal value reflects how hard it is to put the letter on the board (because it’s a rare letter). (Hence rewarding people more for such a move).
Therefore, absent externalities with other letters, a high nominal letter should get a low market price.
The fact that the blank tile has a nominal value of 0 means that it is (indeed) super easy to put on the board. Hence why people are ready to pay a very high price.
What you may look at is whether market price + nominal price = constant, or something like that. I.e. market price inversely related to nominal value.
PS: actually nominal value indicates how hard it’ll be to put the letter, and how rewarding it is. If nominal values are perfectly chosen, the expected return of each letter should be more or less constant. Therefore market price and nominal value should be uncorrelated, and capture only the externalities of the letter.
I’m confusing myself now.
April 4, 2012 at 12:31 am
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