Constitution - a two-person board game

Around the time of the 2020 US presidential election I read a discussion about the electoral college system, where someone claimed that voters in small states have a larger influence on the outcome. This seemed weird to me, but the Wikipedia page on the United States Electoral College contains a similar statement: "individual citizens in states with smaller populations have proportionately more voting power than those in larger states" (together with a reference to an article that doesn't give any evidence for it).

I get that it's more likely that an individual vote (or say a small community) will decide the outcome in a smaller state than in a larger one. But on the other hand the larger state has more electors and is therefore more likely to be pivotal in the electoral college.

It seemed to me that, very roughly, the probability of one vote being pivotal in its state should be inversely proportional to the square root of its population, while the probability of that state in turn deciding the whole election should be directly proportional to its size.

That would indicate that on the contrary, you have more "voting power" if you live in a larger state. But it depends on some assumptions, and it's not super easy to motivate with a simple model.

Anyway, I began thinking "if this was a game and you could choose between influencing voters in a large state like California, or in a small one like Montana, what would be the best move?". Then I scrolled down on that Wikipedia page and saw a cartogram of the distribution of electors. The colored hexagons made it look like a board game.

So I designed one. In the end I thought triangles worked better than hexagons or squares. So here is Constitution, the board game.

This two-person game is played on a board representing a map (admittedly weird looking) of the United States. The board consists of 538 triangular cells divided into 51 regions: the 50 states and the District of Columbia (DC). The cells at the same time represent voters, and on another level, electors.

The central point of DC, where its three cells meet, is the "White House", indicated by a dot in the picture.

The synopsis is that as a presidential candidate, you travel around and campaign, winning voter support and eventually electors. When you get 270 electors, you win the game. If no player gets 270 electors, the winner is whoever won 26 states. Finally, if nobody wins 26 states, you can win by finishing on the White House!

Setup

Each player has a piece of their designated color. The pieces move on the corners of the triangular cells (where up to six of the cells meet). Two such points are taken randomly as starting points (possibly subject to constraints of not being the same or not being collinear). One player chooses one of those two points as their starting point, and the other player makes the first move from the other one. 

Support

At every moment, each cell of the board is either "undecided" or supports one of the two players. Support is represented by a counter or chip of the player's color on the cell. 

The play

The protocol for making a move is as follows:

1. You move your piece. The pieces move in straight lines in any of the six directions defined by the sides of the cells. Pieces can move any distance across water, Canada etc, and can jump over the other piece, but they can only land on a corner of one of the 538 cells. 
2. All undecided cells that have a corner in the point you moved to become your supporters.
   
3. If you thereby obtain support from at least half of the cells in a region (a state or the DC), you win that region. When you win a region other than Maine (ME) or Nebraska (NE), all cells in the region become your supporters, even those that earlier supported your opponent. If you win Maine or Nebraska, you obtain support from all previously undecided cells, but your opponent keeps any support they already had.

An important rule is that you can only make a move if you thereby win support from at least one undecided cell. In other words, you can only move to a point that has an undecided cell in its neighborhood.

Notice:

• If a state has an even number of cells, it suffices to get support from half of them in order to win it. But once you have won the state, you will have support from a strict majority, even in Maine. 
• You can win more than one region in a single move.
• As a consequence of the rules, the two pieces will never occupy the same point, except possibly at the start of the game or after the very first move. And no player can ever return to a point that one of them already visited, except for the starting points. 
• There is no rule specifically prohibiting you from staying where you are, but since you must gather new support on every move, the only time this might be possible is on the very first move.

Electors

When you win a region, the support from that region counts as electors. In Maine and Nebraska, any support immediately counts as electors. Elsewhere, as long as a region is "contested" (no player has support from half the cells) support is only tentative and doesn't count as electors.

End of game

Since the number of undecided cells decreases with every move, eventually one player will run out of move options. When this happens, that player has finished playing, and their piece will remain where it is. Their opponent must then keep playing as long as they have move options (unless the outcome of the game is already clear, see the next section). When no player can make any more moves, the game is over.

Winner

The winning conditions, in order of priority, are as follows:

1. A player with the support of 270 or more electors wins ("in the electoral college"). Electors are cells in regions that have been won by the player, as well as any cells supporting the player in Maine and Nebraska regardless of who, if any player, has won those states. 
   
2. If at the end of the game none of the players has 270 or more electors, a player who has won 26 or more of the 50 states wins the game "in the house". Notice that the DC doesn't count as a state. 
3. If no player wins 270 electors or 26 states, a player whose piece finishes on the White House (the point where the three cells of DC meet) wins the game "in the senate". 

If none of these three conditions applies, the game is drawn. 

Notice:

• It may happen that some regions of the board are left undecided at the end of the game. Conditions 2 and 3 therefore apply not only in case of a 269--269 tie, and a player can very well win the game even if they obtain fewer electors than their opponent. 
• Once a player has 270 electors, the outcome of the game is already clear and the game doesn't strictly need to be finished. The same is true if nobody has 270 electors but one player has won 26 states and the other is already out of moves.
  
• One cannot choose to pass if there is a legal move. In order to apply the winning condition 3, a player therefore has to be "stalemated" at the White House. Reaching it after the opponent has run out of moves isn't enough. 
• Winning according to condition 3 means "occupying" the White House in the sense of residing there, not as in conquering by force.
• Once the outcome of the game is clear, it's appropriate for the losing player to resign, preferably by giving a concession speech. 

Update 2021-07-25:

I had a look at this game again, for no particular reason, and came up with a couple of modifications that I guess will improve it. 

The first is to prohibit moving to a corner of a cell that supports your opponent. This makes it harder in general to move, and it gives another strategic meaning to having support in a given cell. You can take control of a point so that later only you and not your opponent can go there. 

The second is to waive both this and the requirement of having an undecided cell in the neighborhood when it comes to moving to the White House. With the new rule, you can go to the White House from any point aligned with it, regardless of the support situation in the neighborhood, only provided your opponent isn't already there.

I guess we might then after all want to have a rule forbidding you to stay where you are, since otherwise you could just stay at the White House and the game would technically never end.

And another thing: Just because you can't move at the moment no longer means you have finished playing: Maybe your opponent is at the White House now, stopping you from going there, but will have to move on their next turn.

These changes together make winning condition 3 much more interesting. The increased difficulty in moving will tend to make larger areas of the board undecided, increasing the likelihood that none of the players can reach 270 electors or 26 states, and you cannot stop your opponent from later finishing at the White House by moving there yourself.

Also, from a game-playing perspective, the special rules for Nebraska and Maine don't add anything so I guess they should be removed.   

And I might have to update my map.