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State of Nature Game
Game theory can generate useful insights with regards to political science. On example is the state
of nature game which is designed to replicate Hobbesian theory. Hobbes’ philosophy in a nutshell
says that in the state of nature everyone steals from one another and only the strongest survive.
Taking this theory modern political scientists have constructed the state of nature game. The state
Player 1
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
Player 2
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
3, 3
1, 4
4, 1
2, 2
of nature game has two players (we call them player 1 and player 2). Both live in a world with
no government are trying to decide whether or not they should steal or forbear (not steal). In
this society (since there is no government) there is no penalty for stealing. If we were to rank the
possible outcomes we would say that the best possible outcome for player 1 is for player 1 to steal
and player 2 to forbear. Essentially player 1 keeps everything he or she has and then gets some
of what of player 2 has. By doing this type of stepwise analysis we generate the pay−off’s above.
What is the nash equilibrium of this game? If we solve for the nash equilibrium of the state of
Player 1
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
Player 2
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
3, 3
1, 4
4, 1
2,2
nature game we find that both players will steal given this environment. What does this tell us?
It tell’s us that in a world with no government individuals will steal from one another. This is not
good and several philosophers proposed government as a solution to this chaos. What can game
theory tell us about this proposed solution? To find this out we look at the civil society game.
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Civil Society Game
The civil society game differs from the state of nature game in one crucial way. In the civil society
game individuals pay a cost for stealing. Unlike previous games we do not assign this cost a specific
value, but instead note it with the letter “c.” Essentially, this allows to see how the equilibrium
of the game changes as c increases or decreases. We’ll see why this is important in a second. But
first, note the pay−off’s of this game depicted in the game below.. What we will ultimately find
Player 1
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
Player 2
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
3, 3
1, 4 − 𝑐
4 − 𝑐, 1
2 − 𝑐, 2 − 𝑐
is that if the costs are high enough (c>1) individuals in society will prefer to forbear. On the
flipside we find that if the costs are low enough (c<1) individuals will still choose to steal. This is
interesting. It tells us that governments must be strong and capable of providing large punishments
for individuals to prefer forbearing to stealing.
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On a test I am likely to ask...if c=0 what is the equilibrium of the civil society game? How
should you approach this? You should simply take the value of c, subtract it form the pay-offs that
have a c and then solve the game. If c=0 the equilibrium of the game is displayed below. But what
Player 1
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
Player 2
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
3, 3
1, 4
4, 1
2,2
if c=2? If c=2 we have a government capable of imposing significant costs on individuals that steal.
How does this change our equilibrium? Well if do what we did above and subtract 2 from all of the
terms which have a c in them we get the following game which has the following equilibrium1 .
Player 1
3
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
Player 2
𝐹 𝑜𝑟𝑏𝑒𝑎𝑟
𝑆𝑡𝑒𝑎𝑙
3,3
1, 2
2, 1
0, 0
Exit, Voice, and Loyalty Game
This game has two actors. Before the game takes place the state takes something from one of its
citizens. The citizen does not like this (because they lose something) and they are given a decision
about what they want to do. Thus, in the first stage of this game we have a citizen who was
wronged deciding how they should respond. In this game they can respond by, leaving the state,
voicing their opinion (and hoping the state gives them back what they lost), or by remaining loyal.
If the citizen decides to voice their opinion the state can either respond or ignore the citizens. If
the state responds the game terminates (since the citizen gets what they want). If the state ignores
the citizen the citizen gets another choice to either remain loyal or leave the state. Except now
they are paying a cost for initially voicing their opinion. This game is depicted below.
In the question I given you on the test I will give you the values of the parameters you need
to know. So in class we assumed that 𝐸 > 0, that 𝐿 > 1, and that 1 − 𝑐 > 𝐸. Given this we found
one equilibrium. Can you solve the game? If you have any questions please ask.
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Note that we solve these games the same way we solve all other games. If you have any questions see the sheet
from the first game theory class or ask me.
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