Please solve the following problem using a game theory.

Consider a scenario in which two companies, Company A and Company B$,$ are competing for a government contract to supply a specific product. The government has set a maximum budget for the contract, and both companies want to maximize their profits. However, the government will only award the contract to the company with the lowest bid, and the companies have to submit sealed bids simultaneously.

Company A and Company B both have two options: bid high or bid low. If both companies bid low, they will share the contract and split the profits equally. If one company bids low and the other bids high, the company bidding low will win the entire contract and maximize its profits, while the other company will receive nothing. If both companies bid high, neither will win the contract, and they will receive no profits.

The profit structure is as follows $($in millions of dollars$)$:

If both companies bid low: Company A gets $$10$ million, and Company B gets $$10$ million.

If Company A bids low and Company B bids high: Company A gets $$15$ million, and Company B gets $$0.$

If Company A bids high and Company B bids low: Company A gets $$0,$ and Company B gets $$15$ million.

If both companies bid high: Company A gets $$0,$ and Company B gets $$0.$