Destroying Games

So You Want to Break Game Theory

Most people who get into game theory treat it like a set of puzzles with clean answers. The truth is that the models are fragile and the real-world stuff rarely lines up with the textbook Nash equilibria you calculate by hand. I use Destroying Games as a practical framework for taking apart a game-theoretic setup and figuring out where the equilibrium actually falls apart. It is not a single algorithm. It is a workflow. Start by identifying every player, their strategy sets, and the payoff matrix. Then look for the weak spots. The process works like this. Map the normal form. Convert it to an extensive form if the timing matters. Check for dominated strategies. Prune them. Rebuild the reduced tree. See what falls out.

I once spent three weeks debugging a multi-agent auction simulation where the bidders were supposed to be playing a second-price sealed-bid game. The equilibrium analysis from Destroying Games showed me that one of the agents had a hidden cost parameter baked into its reward function. That single overlooked variable made the whole Nash equilibrium collapse under real traffic. The fix was isolating the cost term and adding it as a separate payoff component before rerunning the equilibrium solver.

How to Apply the Destroying Games Workflow

Pull your game into a spreadsheet or a simple Python setup using a library like numpy for payoff matrices and scipy for any optimization subroutines. You do not need anything fancy. A basic linear programming solver will handle most dominance pruning. The first thing you check is dominance. If a strategy always gives a worse payoff no matter what the other players do, remove it. Do this iteratively until no more dominated strategies exist. What remains is your reduced game. Next, check for Nash equilibria in pure strategies. Run a simple best-response check across every cell of the payoff matrix. If you find a pair of strategies where neither player can improve by unilaterally deviating, you have your equilibrium. If not, move to mixed strategies. Solve the system of linear equations that equalizes each player's expected payoff across their mixed strategies.

This part takes time. A 5x5 matrix might take five minutes by hand. A 20x20 matrix with five players usually takes a script. I keep a standard Python routine that accepts payoff tensors and returns both pure and mixed equilibria. It cuts the calculation time down to about two minutes for most scenarios I work with.

Where This Approach Fails

Destroying Games assumes you can fully specify payoffs and strategy spaces. That is a big assumption. In practice, payoff functions are often estimated from data with noise, and strategy spaces may include actions you never thought to model. I have seen setups where introducing a single ambiguous action type completely shifted the equilibrium outcome because the players could bluff in ways the original model did not account for. If your game involves continuous strategy spaces or non-linear payoffs, the standard dominance and best-response checks break down. You need numerical methods or simulation. Tools like OpenSpiel from DeepMind or custom Monte Carlo equilibrium solvers become necessary here.

A Practical Edge Case I Hit

Running a supply-chain negotiation simulation last year, I kept getting unstable equilibrium results. The Destroying Games framework revealed that a participant's private information about their own lead times created an informational asymmetry the model was not designed to handle. Standard symmetric game theory does not cover this cleanly. The workaround was converting the problem into a Bayesian game framework. I introduced type spaces for each player representing their possible lead-time distributions and re-ran the equilibrium analysis. This added about forty percent more computation time but produced results that matched the observed behavior much more closely.

What You Should Know Before Starting

This is not a shortcut to better games. It is a diagnostic tool. Use Destroying Games when you need to understand why a theoretical equilibrium is not showing up in your data or simulation. It will tell you which assumptions are failing. It will not give you a better model unless you are willing to update the model itself based on what you find. If you are just starting out, work with small games first. Two-player, three-strategy setups. Get comfortable reading payoff matrices and spotting dominance. Then scale up. The workflow does not change. Only the computation gets heavier.