Collisions, Combinatorics, and Markov Analysis: `The Balance` Game Design

The Balance is a cooperative card game published by New Venture Games. The objective of the game is to set the top card on eight piles to the same value before the players run out of cards. Players do not know the contents of the other player’s hands and are restricted from communicating on the goal, so part of the challenge of the game is intuiting a shared goal. The game mechanics intersect with variants of the Birthday Paradox and similar probabilistic collision problems. We use probability, combinatorics, Markov analysis, and Monte Carlo simulation to explore the game’s design. We find that there is a sharp difficulty curve in the end game and a transition from players having many opportuntities for exploration to choosing the least-bad option that will not prevent victory. Rules Overview In game theory terms, The Balance is an imperfect information, stochastic, sequential, cooperative (non-zero sum), discrete game. New Venture Games has released a video (for the general audience)…