# Raj

(Rules below the cut)

Raj is my new ‘new favourite game’. I learned about it by reading up on what Boardspace.net has available, and after reading the rules I instantly fell in love. Raj has that essentialness which is shared by certain abstracts. It is the simplest possible expression of a fundamental game mechanic and yet it still playable as an interesting game in itself.

A little historical digging around reveals that Raj is a particular instantiation of Goofspiel or GOPS, a game invented by Merrill Flood while he was a grad student at Princeton in the 30s. It was one of the early examples in game theory.

I roped my step-father into playing a couple rounds with me. We played a three round match; I won the first match, and he won the latter two. We played a somewhat novel variation because we didn’t have the needed cards.

Rules of Raj:

Players: Any number of players, recommended for two or three.

Equipment:

• One set of fifteen bidding cards, numbered 1 to 15, for each player
• Five positive cards numbered -1 to -5
• Ten negative cards numbered 1 to 10

Goal: To collect the largest value in prizes by bidding well.

Play: The positive and negative cards are shuffled together and placed face down. This is the set of prize cards to be won. The first prize card is then flipped over and bid upon. Each player secretly selects one of their bidding cards. One all players have selected a card, all the bids are revealed. If the prize card was a positive card, the highest bidder keeps the prize. If the card was a negative card, the lowest bidder keeps the prize.

The tie problem: I can’t make out what one should do if there is a tie. Everything I’ve read online seems to suggest different rules for playing with two and playing with more than two. Here are my personal thoughts on the matter which try to collapse these two rules into one unified rule.

If there is a tie for the prize, then the prize goes to the person who would take the prize if the tied players hadn’t played. If there is a tie amongst the second tier prize takers, it goes to the person who would have won if the first and second tier hadn’t played. This tier process should continue until either: there is a unique winner on some tier, or there is no remaining players to form the next tier. If there are no players for the next tier, the contested prize card is set aside and its value is added to the value of the next prize card. Play proceeds as normal.

Another, better, way of saying this: cancel all matching bids. Now reconsider who takes the prize. If all the bids have canceled, the contested prize card is set aside and its value is added to the value of the next prize card. Play proceeds as normal.

Note: There is a subtlety here that I ran into when trying out Raj — sometimes the value of a prize obtained by carrying over a tied prize can be zero. This makes the rule about which bid wins the prize somewhat ambiguous. I’ve adopted the convention that the rule above refers to the positivity or negativity of the last card added to a collective prize.

Once it has been settled who won the prize, all the bid cards are discarded and the player who won the prize keeps it. Once all the prize cards have been played all the players total up the values of their prize cards. The player with the highest total wins.

When my step father and I played we didn’t have nearly enough card games around to improvise a deck, and so we came up with the following pencil and paper version of the game. For prize cards we used playing cards: taking ace through ten of hearts for the positive cards and the ace through five of spaces for the negative cards. For everything else we used pencil and paper.

We each had a piece of paper for writing down our bids and a communal piece of paper for keeping track of what had been played (see below) and the score. I’m not much of a card counter, but my card-shark of a step-father sure is, and so we kept tally of what had been played, so that I didn’t have to. For a more astute card counter, one would only need a score sheet and bid card.

$\begin{array}{cc|c|cc} Bids & & Prizes & Score &\\ \hline A &B & &A &B\\ \hline 15 &15 &10 & &\\ 14 &14 &9 & &\\ 13 &13 &8 & &\\ 12 &12 &7 & &\\ 11 &11 &6 & &\\ 10 &10 &5 & &\\ 9 &9 &4 & &\\ 8 &8 &3 & &\\ 7 &7 &2 & &\\ 6 &6 &1 & &\\ 5 &5 & & &\\ 4 &4 &-1 & &\\ 3 &3 &-2 & &\\ 2 &2 &-3 & &\\ 1 &1 &-4 & &\\ & &-5 & &\\ \end{array}$

Instead of playing bid cards to bid, we wrote them down our bids behind a screen. Jokers proved to be thematically nice for this. All in all, it worked rather well.