I am currently looking at the nakamoto coefficient for tezos. The nakamoto index is the number of participants needed to get the absolute majority in a system, in bitcoin for example this would be 51%, i.e. the number of participants needed to get 51% of the stakes in the system would be the nakamoto index. For certain blockchains such as algorand one needs 2/3 of the participants in order for the system to achieve consensus, therefore the nakamoto index would be the number of participants needed to have more than 1/3 of the total stakes. So how many participants are needed for the system to act maliciously in tezos? (one third/one half or another number)
For the current consensus protocol, Emmy*, the attacker needs at least half of the total stake, since it's a Nakamoto-style algorithm (like Bitcoin and Ouroboros).
For the consensus algorithm of the next protocol proposal, Tenderbake, the attacker needs at least a third of the stake, since it's a "classic-style" algorithm (like PBFT, Tendermint, Casper). (The attacker might need slightly less, say 31%, because validators are a randomly sampled subset of all possible participants, and the attacker may be lucky sometimes.)