Graphs of Net Benefits, Costs, and Payoffs for Fix(x) and 'Var'

 

Graphs For Fix(x=m):

The first curve is very simple -- it just shows that we assume that costs increase linearly (with a slope of 1) over time.

This next graph once again shows the costs that are paid (blue line, same as above) and adds resource value (violet) and net benefit (V-m). Resource value is, of course, constant and independent of cost. Therefore in longer contests with greater costs, the net resource value (i.e. the net gain from winning) eventually becomes negative:

The final graph (below) compares costs, resource value, net benefit, and payoffs in ties. As before, costs are a linear slope = 1 function of time, resource value is constant and net benefit is (V-m).

Now:

Thus, overall, E(fix(x=m), fix(x=m)) is the sum of these: 0.5(V-m) - 0.5m = 0.5V-m.

This is graphed as the red line below:


For the Mixed Strategy 'Var':

Before Continuing, Let's Be Certain
That We Understand This Graph. 

Axes:

  • The Y axis is either cost, benefit or payoff; all are in the same units as V. As usual we show costs as positive values (our convention then is to subtract costs from benefits. By the same token, net benefit in winning (yellow line) can be positive (if V > cost of winning V), negative (if V < cost of winning V) or neither.
  • The X axis is the cost x.

It is extremely important that you realize that all of the curves above are lifetime or theoretical curves. Thus, the payoff curve (magenta) is the solution giving the lifetime payoff to a var strategist when pitted against a all possible cost fix(x) strategist if V=1.0.

Thus, the particular fixed strategist that var is playing is given as the x axis value. For example: E(var, fix(0.5V)) = 0.212V; E(var, fix(0.2 V)) = 0.64V, etc.

Now, let's see what is interesting about this graph:

 Press here for a "floating window"
containing the plot we are discussing

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Recall that 'var' tends to quit most contests at relatively low costs (press here to review the plot of the chance that 'var' will play to some given cost (given its opponent has not quit)).

 Press here for a "floating window"
containing the plot we are discussing

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