The world chess championship has seen an increasing number of draws since the wide$-$spread use of computer

preparation. One potential rule change that has been discussed is to give the world champion draw odds: if

the match ends in a draw, the world champion retains the title. The motivation for the proposed rule change

is that this forces the challenger to adopt a more risky play style that leads to more decisive outcomes.

To analyze whether such a rule change has the desired effect, we need to consider how rational players react

to the rule change. Let us model the interaction as a game, in which both players can adopt either a risky or

a safe strategy. Both players are skilled enough to ensure a draw if they choose the safe strategy. If they both

play a risky strategy, the result is decisive: the challenger wins with probability p and the world champion

wins with probability $1$ p$.$ A player gets a utility of $1$ for winning the title and $0$ otherwise.

a$.$ Write down the payoff matrix with the draw odds rule in place.

b$.$ Find all Nash equilibria.

c$.$ Do you believe draw odds will successfully reduce the number of draws we observe? Explain.