Yes, you can play on the LAN.
For the player who will host the game, click on Multiplayer.
In the lobby screen, click on LAN and choose Yes when it asks if you want to run the LAN server. The server address will change to be your local IP address. Then create a game as normal.
The guest then just has to change the value in the server address entry field to be your IP address (hit ENTER to make sure that it gets updated). The guest should now be able to see and join your game.
It is possible to activate The Political Machine 2008 on multiple machines, but you only get a certain number within a set timeframe, so don't just give your serial number to your buddy or you may go to update one day and not have any activations left because he gave it out to all of his friends too. For offline activation instructions, please read this knowledgebase article:
https://esupport.stardock.com/index.php?_m=knowledgebase&_a=viewarticle&kbarticleid=447
If you exceeed your number of activations, you will need to contact
[email protected]