Previously actions map to states. Not now.
Previously we modelled utility as a function of variables in control of the agent, or constants. We now add another type of term: variables controlled by other agents.
Consider a simple pair of agents:
\(u_a=f_a(x_a,y_a)\)
\({actions_a}=\{x_a,y_a\}\)
\(u_b=f_b(x_b,y_b)\)
\({actions_b}=\{x_b,y_b\}\)
Each agent’s decision does not affect the other agent. Consider now a utility function:
\(u_a=f_a(x_a,y_a)\)
\({actions_a}=\{{x offer}; {y offer}\}\)
\(u_b=f_b(x_b,y_b)\)
\({actions_b}=\{accept; reject\}\)
Where \(a\) offers a trade to \(b\) and \(b\) accepts or rejects. This is an example of a sequential game. There are many types of game, with differing implications.
Economic agents face options from some set. This could be consumption choices, numbers of hours to work, or how much capital to invest in at a factory.
Consider the prisoner’s dilemma game: table.table.table-bordered thead tr th th Silent th Tell tbody tr td Silent td (5,5) td (10,0) tr td Tell td (0,10) td (8,8)
In this game we have two agents who simultaneously choose
Let’s compare the decision to “tell” to the decision to be “silent”.
No matter what the other agent does, you are always better off choosing “tell”. As a result we say the strategy “tell” strictly dominates “silent”.
If under some circumstances the agent is indifferent to the strategy and another, then the strategy only weakly dominates.
So one way to solve a game is to choose dominating strategies. However an agent may not have strictly dominating strategies. Another method it to rule out strategies. If one strategy is strictly dominated for an agent, we can rule out them choosing it. This may reveal strategies which are dominant one actions of another agent can be ruled out.
If after iterations of this process we are left with only one strategy for each agent, we say this is a Von Neumann solution, an analytic solution.
But what if there are still multiple options? Consider
table.table.table-bordered thead tr th th Opera th Tell tbody tr td Opera td (10,5) td (0,0) tr td Football td (0,0) td (5,10)
And
table.table.table-bordered thead tr th th Rock th Paper th Scissors tbody tr td Rock td (0,0) td (-1,1) td (1,-1) tr td Paper td (1,-1) td (0,0) td (-1,1) tr td Scissors td (-1,1) td (1,-1) td (0,0)
In both of these there is no strategy which is always better to follow, even weakly. But these games are very different. In the former, if agents agree to both got to football, or both to opera, neither would be better off by defecting. In the second example there is no such “Nash equilibrium”.
This is relevant for considering how to expand the game. In the former example a couple can talk to each other and coordinate actions. For example one agent could commit to going to the football, and the other agent would rationally join.
In the latter no such coordination is beneficial.
In the context of the game, the player can instead of choosing a pure strategy such as “rock”, which may not always be appropriate, choose a mixed strategy.
For example a player could choose each of the \(3\) moves \(\dfrac{1}{3}\) of the time.