Spoiler: This article reveals parts of the story of Liar Game

Rules

The last game of Liar Game oppose 4 kingdoms. The rules are simple:

• There is only one winner.
• Each team starts the game with 100 life points.
• Each round, a kingdom can choose to attack between 0 and 3 opponents and defend against 0 to 3 opponents, for a maximum number of 3 actions per round.
• Each action costs 1 life point
• Not defending against an attack costs 3 life points.

One on one

At first, I thought that the game could be considered as 3 parallel one on one battles (spoiler: I was wrong).

In a one on one battle, a round can be considered as two simultaneous sub-battle with one attacker and one defender. According to the actions, from the point of view of the attacker (spoiler: wrong again), the score difference will be:

• no attack, no defence: +0 points
• attack, no defence: +3 -1 = +2 points
• no attack, defence: +1 points
• attack, defence: -1 +1 = 0 points

Let say that the attacker will attack A% of the time (randomly), and the defender will defend D% (randomly). I consider random actions because it makes in practice impossible to predict a pattern. Then the score difference will be:

(1-A)*D + 2*A*(1-D) = 2*A + D – 3*A*D

I realized that there is a strategy for both attacker and defender that minimises the point loss (2/3 point for each round), whatever is the strategy of the opponent

• for the attacker, attack 1/3 of the time
• for the defender, defend 2/3 of the time

If the opponent can detect another frequency, then he will adapt his own frequency to get a greater gain (which can be countered by another frequency; the winner will be the one which will be the faster to detect and to adapt itself).

4-way battle

It turned out that I was partially wrong. I forgot about several things:

• A kingdom can only do 3 actions for a given round (so you cannot defend at 66%, attack at 33% and attack even more if you detect a defence pattern).
• In a 4-way battle, when a kingdom can land a successful attack, two other kingdoms get the benefits of the attack, but only the attacker is losing an action point! If you manage to land a successful attack on each opponent, you will lose 3 points and each opponent will also lose 3 points (and it’s the best scenario!). So one way to see the goal of the game is to try to minimize the point loss (doing nothing is a good start), and the best interest for a kingdom is not attacking at all and let the other attack his opponents.
• If two kingdoms team up, they will save 1 defence point (and 1 attack point) for each round because of their mutual trust. So making alliances (and betraying) is maybe the centre of the game.

A friend suggested me that a good strategy may be to propose an alliance, create a advantage compared to the other twos… and betray the ally before he does, and try to keep the 3-points advantage of a successful attack (because the kingdom having the largest number of points at any time is the most likely to win, thanks to the 2/3 defence strategy).

I think that it is an interesting game. Even if the game is balanced at the beginning, at some point it will not be anymore, and the team with a lower number of points should team up against the one(s) with the greater number.