🔎 Split or Steal

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

Roleplaying as Ron to find Ron’s underlines (best responses)

If Cathy plays Split, what is Ron’s best response?

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

If Cathy plays Steal, what is Ron’s best response?

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

Roleplaying as Cathy to find Cathy’s underlines (best responses)

If Ron plays Split, what is Cathy’s best response?

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

If Ron plays Steal, what is Cathy’s best response?

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

Summarizing all of this in a single game grid, we get:

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

There are three Nash Equilibria:

• →(Ron → Steal, Cathy → Split)
• (Ron → Steal, Cathy → Steal)
• (Ron → Split, Cathy → Steal)

Why is the following strategy profile a Nash Equilibrium for poor Cathy:

→(Ron → Steal, Cathy → Split)

It is a NE because Cathy can’t do any better than she already is.

Right now, we are in the bottom left box:

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

You might think that because Cathy has a payoff of $0, that this wouldn’t be an equilibrium for her. However, we will find that she can’t do anything at all to get a payoff of more than $0. She’s stuck.

Cathy only controls the column that we are in. She can choose whether she steals, but not whether Ron steals.

Her only option to deviate from this Nash Equilibrium is the decide to steal herself. If she decides to steal herself, she would get a payoff of $0. That’s not an improvement. She might be deviating, but is isn’t profitable.

Economists will often define NE as “A strategy profile (a list of what all players are doing), where no player can profitably deviate.” We have just verified that even though Cathy may be unhappy, she can not profitably deviate.

Because Cathy can’t control Ron, it is a best response for her to play Split when Ron plays Steam. In other words, she can’t profitably deviate.

In the split or steal game, for both players, it is a “weakly dominant strategy” to steal. This means that it is never worse to steal, but it isn’t always better to steal. For regular dominant strategy, just “not being worse” isn’t good enough. It must actually better.)

For example, let’s confirm that stealing is weakly dominant for Ron.

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

If cathy is going to split, is it better for Ron to Steal? YES.

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

If Cathy is going to steal, is it better for Ron to Steal?

	Cathy → Split	Cathy → Steal
Ron → Split	R=$20k, C=$20k	R=$0, C=$40K
Ron → Steal	R=$40K, C=$0	R=$0, C=$0

It’s definitely not worse!!

Because it’s not worse for Ron to choose steal, independent of what Cathy does, Stealing is a weakly dominant strategy for Ron.