What is the Discouragement Effect?
The discouragement effect is when the future consequences of winning or losing the current contest leads to decreased effort Konrad (2012). This is due the future contest reducing the overall value of winning. Think about a tennis match with 3 sets. If a player loses the first set then to win the entire match they must win 2 sets compared to the other player only needing to win 1 match. Therefore they have a harder path to victory than the other player and have an incentives to reduce their effort.
Example
Now let us work through an example. We will continue with our tennis exposition and place the contest as the final at the Australian Open. To win a player must be the first player to win 3 sets.
\[\begin{equation} playerPayoff = probabilityOfWinning * (prizeOfWinning - prizeOfLosing) - costOfTrying \end{equation}\]Remember that the probabilityOfWinning is the output of function called the Contest Success Function with takes both players efforts as inputs and the costOfTrying is the function with the input of the player’s effort. To make this less dry as possible lets say the grand final is between Roger and Nadal and that Roger is winnign 2 sets to 1. This means that if Roger wins the next set we wins the championship but id Nadal wins then both players have 2 sets each and the final set winner decides the total winner.
Now if Roger wins he gains the netPrize, which we will give the value of 1. If Nadal wins then both players are tied at 2 sets each. To work out the valuation of being in this state we need to introduce the Tullock model for the CSF.
\[\begin{equation} porbabilityOfWinning_{Roger} = \frac{effort_{Roger}}{effort_{Roger} + effort_{Nadal}} \end{equation}\]Pretty neat huh. Now to simplify let us make the costOfTrying to be unity ie
\[\begin{equation} costOfTrying = effort \end{equation}\]Now we have the tools to work out the payoffs for both Roger and Nadal when they are tied at 2 sets each. If either player wins they gain the prize, which we have said is 1.
Now we need to solve the optimisation problem.
\[\begin{equation} \frac{\partial payoffRoger}{\partial effortRoger} = \frac{\partial payoffNadal}{\partial effortNadal} \end{equation}\] \[\begin{equation} \frac{effortNadal}{(effortNadal + effortRoger)^2} * 1 - 1 = \frac{effortroger}{(effortNadal + effortRoger)^2} * 1 - 1 \end{equation}\]Simplifying we achieve \(effortNadal = effortRoger\)
Now if we put this back in our original equation
\[\begin{equation} payoff = \frac{effort}{2*effort} * 1 - effort \end{equation}\]Simplify to
\[\begin{equation} payoff = \frac{1}{2} * 1 - effort \end{equation}\]Therefore effort and payoff are equal to 0.25.
Now this means that Roger faces winning and gaining 1 or losing and gaining 0.25 and Nadal facing winning to gain 0.25 and losing to gain 0. Therefore the value for winning to Roger is 0.75 and for Nadal it is 0.25. This causes asymmetries in winning valuations.
Now lets worked out how hard the players play at 2 sets to 1.
\[\begin{equation} payoffRoger = \frac{effortRoger}{effortRoger + effortNadal} * 0.75 - effortRoger \end{equation}\] \[\begin{equation} payoffNadal = \frac{effortNadal}{effortRoger + effortNadal} * 0.25 - effortNadal \end{equation}\]Now we need to work out:
\[\begin{equation} \frac{\partial payoffRoger}{\partial effortRoger} = \frac{\partial payoffNadal}{\partial effortNadal} \end{equation}\] \[\begin{equation} \frac{effortNadal}{(effortNadal + effortRoger)^2} * 0.75 - 1 = \frac{effortRoger}{(effortNadal + effortRoger)^2} * 0.25 - 1 \end{equation}\]so \(effortNadal * 0.75 = effortRoger * 0.25\)
Which simplifies to \(3*effortNadal = effortRoger\)
Now subbing these back into our equations we get
\[\begin{equation} \frac{effortNadal}{(effortNadal + 3*effortNadal)^2} * 0.75 = 1 \end{equation}\]This simplifies to give an effortNadal of 0.046875.
\[\begin{equation} \frac{effortRoger}{(0.333*effortRoger + effortRoger)^2} * 0.25 = 1 \end{equation}\]This gives an effortRoger of 0.140625 which also satifies out equality given before.
\(0.046875 + 0.140625 = 0.1875 < 0.5\) Therefore our total effort has been reduced in this contest.
Further Reading
Give this one a go Konrad (2012).
Konrad, Kai A. 2012. “Dynamic Contests and the Discouragement Effect.” Revue d’économie Politique 122 (2). Dalloz: 233–56.