Ridere, ludere, hoc est vivere.

Monday, September 10, 2018

Bidding and game theory

I have been thinking about the game theory construct for the share auction in Chicago Express (designer Harry Wu, publisher Queen). The question isn't only one of absolute valuation but also one of the interactive decision-making in the auction. That thought led to consideration of the auction as a game-theory problem.

In CE, players don't own railroads outright. Instead they buy shares of railroad companies at auction in anticipation of earning income from those shares later in the game. Let's simplify the problem and suppose that Anne initiates an auction for a share of Pennsylvania Railroad. Let's assume that both Anne and Bob value a share at $3. Let's further assume that Anne and Bob each have $5, and that the minimum bid with which Anne can open the auction is $1. (These aren't realistic values in CE but serve to illustrate this game theory case.)

We can look at this auction as a game with sequential moves, which I wrote about during our study of Games of Strategy by Dixit, Skeath, and Reilly. The auction is a fairly simple case of such a game that we can network fairly easily. Let's simplify the game one step further and say than on a turn, a player can either pass, or raise the opponent's bid by $1. We can create a network diagram showing the progress of the game from left to right, where each node is a player's decision point, and each arrow is the result of a decision at that point - whether the end of the game and a resultant score, or progress to an opponent's decision point. If Anne wins the auction, the value of the game is represented as the value of the share ($3) less what Anne paid for it - positive if she made a profit, negative if she paid too much for it. If Bob wins the auction, the value of the game is the opposite - how much he paid less the value of the share - so that Bob prefers a negative result, and Anne prefers a positive result.

Anne must bid 1 as her first move. Bob can either bid 2 or pass. If he passes, the value of the game is $3 less Anne's $1 bid, or +$2. If Bob bids 2, now Anne can either bid 3 or pass. If she passes, the value of the game is the $2 that Bob bid, less the $3 value of the share, for a result of -$1. The decision tree continues from left to right, until Anne bids $5, which ends the game because Bob does not have more than $5 to bid.

Now we can start pruning branches from the network that the players are not motivated to follow. If Bob bids $4, Anne will not bid $5 because she would lose $2; she would rather pass and see her opponent lose $1 instead (reflected as a +$1 value in the game). As we work our way backward, Bob realizes that if he bids $4, he will lose $1 because he knows Anne is motivated to pass, so he will pass instead. That means that Anne knows she will break even if she bids $3 because that is a better result than passing after Bob bids $2 and letting him gain a $1 profit. Bob likewise realizes that he is better off bidding $2 than passing and letting Anne gain a $2 profit. After pruning, the game is solved with the result that Anne will break even when she buys the share for $3.

Now, the game is actually a little more complicated, because players can raise their opponent by more than just one dollar. For Anne's first move, she can bid anywhere from $1 to $5. Bob's next move is either to pass (allowing Anne to take the share for the amount she bid) or to bid a higher value than Anne, but not more than $5. The result is a somewhat more elaborate decision network.

After going through the same pruning exercise, we find that the outcome of the game is that one player or the other will end up buying the share at its value of $3 for no profit and no loss. So at any point in the game, either player can either bid $3 to take the share, or bid $2 to force their opponent to take it. Since Anne is the first bidder, the decision is hers to make; if she bids $1, she is deferring that decision to her opponent.

The game gets a little more interesting if the share has different value to the two players. If the share is worth $2 to Anne but $4 to Bob, Anne likely to bid $2 or $3. If she bids $2, Bob will buy it for $3 at a profit of $1; if she bids $3, Bob will pass and allow her to lose $1.

The game is even more interesting if the players perceive the value of the share differently. If Anne believes it is worth $2 regardless of who owns it, and Bob believes it is worth $4, then Anne will bid $1 or $2, and Bob will buy it for $2 or $3.

If the players misjudge how their opponents value the share, then each is essentially playing the game on a different decision tree, because they believe their opponent's decisions will be based on a different payout result. Suppose Anne values the share at $2 and believes Bob values it at $2 as well, but Bob values it at $4 and believes Anne does as well. He may bid $3 on the share to "force" Anne to bid $4, when instead she will pass and leave him with a profit of $1. If Bob had realized Anne valued the share at $2, he would have bid $2 to induce her to pass and gained a profit of $2.

This observation leads to what Dr. Aaron Honsowetz of the Dr. Wictz design team described to me as the Winner's Curse, the phenomenon by which the person who most highly estimates the value of an auctioned item will bid the most and thereby win the item. Assuming that the actual value of the item is closer to the average estimate of the population of potential buyers, the winner is therefore likely to have overpaid for it.

The motivation for this whole line of thinking was to consider bidding strategy in an auction game. I conclude that if a player doesn't know how his or her opponents value the share in question, it makes sense to bid or raise the minimum in the hope that all the opponents will drop out before the bid exceeds one's own valuation. But if that happens, the player is left to wonder if they've been stuck with a share that is worth less than they think. On the other hand, if the value of a share is relatively clear to all the players, then the logical play is either to bid the value of the share (to take control of it) or just a dollar less (to force an opponent to take it at essentially no profit).

Sometimes I will underbid for a share by an amount equal to the number of players; e.g. if I value the share at $10 in a four-player game, I will bid $6. If every player raises the bid by $1, I'll still have the opportunity to buy it for $10 - or less, if anyone dropped out of the auction before my turn came up again. But that strategy can backfire if someone else jumps to a $10 bid first. So as is always the case in these games, reading the opponents is as important as valuation.

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