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Studying for a test? Prepare with these 2 lessons on Game theory and Nash equilibrium.

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# Game theory of cheating firms

Video transcript

Male Voice: In the last video we saw how there could be an industry
that has two firms, a duopoly, and if those
two firms coordinate they could behave as a
monopolist and they could optimize their collective economic profit. In the last video we saw that would happen when they produced 50 units per period, and they could split
it, assuming these were two identical firms, by
each producing half of it. In the case of the last
video, it was 250 units per firm. Then we saw that
there was an incentive to cheat; that by producing extra units, from a market's point
of view, the marginal economic, or the economic profit on those incremental units would
be negative, so the whole economic profit would shrink a little bit as you produced units beyond that, but the cheater would get a bigger chunk of those units, or the
bigger chunk of that economic profit. The
cheater could actually gain, go from $250 per time period to $280, and it would be all at the
expense of the non-cheater, and then some, who would lose even more than what the cheater gained. Obviously who was
initially the non-cheater has an incentive now to cheat, and they'll both keep increasing,
they'll both keep increasing production so that if they wanted to keep doing this one-upmanship. They both have the incentive to keep
going assuming that they don't hold to their cartel agreement until you get to a
quantity where there's no economic profit left.
Right over here, the way I've drawn it, the demand curve intersects the average total cost
curve right over here, and there's no economic profit left. We're producing a good quantity. It looks like it's
about 75 units combined; 75 units for the whole market. But at this point, the market price is equal to the average total cost, and so there's no economic profit per unit on average. What I want
to do is think about this in kind of a Game theoretic way. Let's look at a bunch of states. This is the optimal state
that we are starting off in. You can actually call it
the Pareto optimal state, named after Vilfredo Pareto. All it means is that's
the state where there's no other state where you can make someone better off without making
the other person worse off. Any of the states here, there are states, for example, where blue is better off. For example, in this state right over here blue is better off,
but green is worse off. So that's why it's
called Pareto optimality. Now, what I want to think
about is how these characters will change their state
due to their incentives. Then we'll talk a little bit
about Nash Equilibrium as well. On this axis, up here,
lets' say this is one of the competitors. This
is where they produce 25, and let's say on the ultimate cheating quantity of 75, and this
is somewhat close to the market, or that is
the equilibrium quantity if this was perfect competition, they produce half of that, so this is them producing 37.5 units. As we
go from 25 to 37.5 units, they are cheating more.
This is more cheating and over here, this was no cheating. We can do the same thing
for the blue player. I'll write them as B.
This is them producing 25. This is them producing
37.5. As we go up and up and up, they are cheating more. This is a lot of cheating,
or more cheating. To think of it in a Game theoretical way, this is the Pareto optimal
state right over here. It's optimal in many ways. This is they've maximized
the total economic profit here. There's
no other state that one person would benefit without
making the other worse. Now, let's think about whether this is a Nash equilibrium. Let's remind oursleves what Nash equilibrium was. This was a state where
holding all the other players constant, so in
this case there's only one other player, a player can't
gain by changing strategy. In this case, changing
strategy is changing your output. Let's see if
that is true of this state right over here. Well,
let's hold A constant. If A is constant, we're
in this column right over here. Is there something B can do, is there change or strategy B can do, that would allow B to gain? Sure. B can increase production. That's what we saw in the last video. We would go from this bottom right state to one right above it.
Now B's economic profit is 280, A's is 200. The pie has shrunk, but B has got a larger chunk of it. That was not a Nash equilibrium. There is, holding all others constant, there is a player that can gain
by changing their strategy. The Nash equilibrium
definition, just to make sure, they say it's a state where holding others constant no player can
gain by changing strategy. We just showed that at
least one player can gain by changing strategy
holding others constant. The same would be true if we
went the other way around. If we held B constant at 25, A could gain by changing his strategy,
could go right over there. This is not a Nash equilibrium. Then regardless of what state we go to, if we go to this state, it's still not a Nash Equilibrium. If we hold A constant, B could improve by
increasing his production; or if we hold B constant, then A can still improve by cheating even more. None of these are Nash equilibriums. From any one of these states,
if you hold A constant, B could produce more; or
if you hold B constant, A could produce more and get some gain. Over here, A's going from
130 to 160 and getting some gain. You can imagine
this keeps happening incrementally. They keep
producing more and more and more. We kind of go
there, then we go there, then maybe we go there, then we go there. Then maybe A cheats some more, then B cheats some more, then A
cheats a little bit more, B cheats a little bit more,
maybe a little bit more past that, then A cheats
a little bit more. The whole time the whole
economic profit pie, which is the sum of A
and B, is getting smaller and smaller until finally A finally cheats and they're at zero economic profit. Now let's think about whether this is a Nash equilibrium.
Clearly, they won't want to move backwards. If you hold A constant, B would not want to
move down. Then he would lose economic profit. That doesn't work. He doesn't gain by doing
that. If you hold B constant, A wouldn't want
to move to the right. A would also lose economic profit. Now you might say what if
they produced beyond 37.5? Why can't they keep producing
and go beyond there? Holding A constant, if B were
to produce more than 37.5 from this state right
over here, then the total pie will get negative
and it doesn't matter if B's getting a larger or
smaller chunk of that pie. B's chunk is going to be negative. He's going to drive down
the price even more. You can see it over here. If they increase quantity beyond this market quantity of 75, 37.5 each, if we go beyond that,
the price that they would be selling at, at that
quantity over there, is lower than the average total cost. You're going to be, the total economic, the average economic profit per unit is going to be negative.
There will be a total of negative economic profit. Neither of them will want to produce more from this state either. All of a sudden in this topless state,
holding others constant; if you hold A constant, B can't gain by changing his strategy,
and if you hold B constant A can't gain by changing his strategy, so we are, up here, in a Nash equilibrium. This is a Nash equilibrium. Like the prisoner's dilemma, it was not the optimal state. The
optimal state was here, but because they both wanted to cheat, they both wanted to do this one-upmanship, they both broke their contracts, they could end up in this state over here. But this state is stable. There's nothing holding
the other party equal. There's nothing that they could
do to change, to optimize. What they could do, and
this is not what Nash applies to, they could say okay, we've been really ruining
each others' business. Let's go coordinate again
and I'm going to decrease production if you decrease production. That is not, and they
could maybe try to go back to this state, and that does
not mean that this is not a Nash equilibrium because
by coordinating again we're not holding the others constant. We're saying I'm changing
my strategy while you're changing your strategy. Maybe only through another agreement
they could go over here. That still doesn't mean that this is not a Nash equilibrium. This
is a Nash equilibrium. If there's no coordination,
if you hold one player constant, the other player
cannot change their strategy, or change their production, for a gain.