H. G. Muller wrote on Thu, Sep 27, 2018 09:53 PM UTC:
I think this is where Betza's 'leveling effect' comes in. You can use a piece in two ways: (1) avoid trades for a nearly equivalent opponent piece; (2) don't care about such trading. In the trade-avoiding strategy (1), the opponent's counterpart will interdict access to the squares it attacks, as going there would give him the opportunity to trade. This limits the use you can make of the piece, thus depressing its effective value. In general, stronger pieces lose value due to the presence of opponent weaker pieces that they have to avoid 1-for-1 trading with.
If the value was close to start with, the value depreciation caused by adopting a trade-avoiding strategy can be larger than the intrinsic difference. In that case you would be better of using strategy (2). But there the fate of the piece is to be traded, which makes them effectively equal in value, as any difference will evaporate with the trade. So pieces nearly equal in value will see their value pulled towards each other when they oppose each other, until it gets exactly the same. I think this is pretty much the case for a Knight and a lone Bishop on 8x8. If the intrinsic value of the Bishop was somehow increased compared to the Knight, initially you would not benefit from it. Because you would have to 'sacrifice' that extra intrinsic value by limiting the use of the Bishop by stricter trade-avoiding.
I think this is where Betza's 'leveling effect' comes in. You can use a piece in two ways: (1) avoid trades for a nearly equivalent opponent piece; (2) don't care about such trading. In the trade-avoiding strategy (1), the opponent's counterpart will interdict access to the squares it attacks, as going there would give him the opportunity to trade. This limits the use you can make of the piece, thus depressing its effective value. In general, stronger pieces lose value due to the presence of opponent weaker pieces that they have to avoid 1-for-1 trading with.
If the value was close to start with, the value depreciation caused by adopting a trade-avoiding strategy can be larger than the intrinsic difference. In that case you would be better of using strategy (2). But there the fate of the piece is to be traded, which makes them effectively equal in value, as any difference will evaporate with the trade. So pieces nearly equal in value will see their value pulled towards each other when they oppose each other, until it gets exactly the same. I think this is pretty much the case for a Knight and a lone Bishop on 8x8. If the intrinsic value of the Bishop was somehow increased compared to the Knight, initially you would not benefit from it. Because you would have to 'sacrifice' that extra intrinsic value by limiting the use of the Bishop by stricter trade-avoiding.