The nothing-at-stake problem

When Casper with PoS is implemented, participants will be able to push blocks forward by placing a bet on them if they believe they are valid while getting rewarded if they choose the valid one. But, what happens if they decide to vote on a malicious block? Nothing stops them for doing so, since they don't lose anything when betting on a malicious block. This is known as the nothing-at-stake problem.

These miners will be able to bet on every block they see for a possible reward.

One possible solution to this issue is to punish those that are constantly betting on the wrong block. So, if a miner makes 10 bad calls, he will lose a portion of his stake. However, this only happens if the majority of the network decides to vote for the wrong blocks, if the miners are distributed enough it would be very hard to agree on the wrong choice.