No universal adoption would not end small wars.
The reason is that the use or lack of use of nuclear weapons is down to a carefull balancing by either power on it's ability to win or prevent defeat. If everyone has nukes then it depends on who you are allied with do you and all you allies have more nukes than you enemies. If you do not have more nukes than you enemy then you need to be sure you enemies will not attack, if you cant be sure then you may
need to attack in order to show you would attack your enemies allies if they respond.
I effect you need to make you enemy understand you will respond, if they do not understand then the likelyhood of miscalculation increases.
Also the less nuclear weapons involved the more likely there use, if we all have only have 10 nukes then even a small countries might think they can gain advantage from there use.
The calculation works like this
The more nukes you have the greater the chance of use.
The less nukes you have the greater the chance of use.
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