Finding a solution is one thing. But how hard is it to find that solution?
In game theory, the solution is often a Nash equilibrium. But computing it can be tricky.
In a game, a Nash equilibrium is a set of strategies. No player wants to change their strategy if the others stay the same.
For example:
Finding that sweet spot is easy in small games. But it gets hard fast.
Games with many players and choices are complicated.
There may be many equilibria. Or none in pure strategies.
So we need to look for mixed strategies. That means players randomize.
Computers struggle with these cases. They must check many possibilities.
It tells us how hard a problem is to solve using a computer.
Some problems are easy. We can solve them in polynomial time.
Others are hard. They may take too long, even on fast machines.
This is an important result.
Finding a Nash equilibrium is in a class called PPAD.
PPAD stands for "Polynomial Parity Argument, Directed version."
It sounds scary, but here’s the idea:
Nash equilibrium is one of them. We know one always exists (thanks to Nash!). But finding it is hard.
In fact, it’s PPAD-complete. That means it’s one of the hardest problems in this class.
Companies compete for ad space on Google.
Each company has a strategy: how much to bid, when to show the ad, etc.
Google wants to compute an equilibrium. That way, ads are shown fairly and profitably.
But computing this is not easy. It may require solving a PPAD-complete problem.
Traders make choices based on others' moves.
They want to reach a stable state. No one wants to trade differently.
Finding this stable point is like computing an equilibrium. And it’s complex.
Yes! In practice, we use algorithms that find approximate equilibria.
They may not be perfect. But they’re close enough.
And they run faster. This is helpful for real-time systems.
So next time you play a strategy game or bid online, remember: even the computer might struggle to find the best move! 😄