## Finding the winning strategy on the Grundy’s game

Draw the game tree for Grundy’s game: Two players have in front of them a single pile of objects say stack of 9 pennies . The first players divides the stack in to two unequal pile , second player does the same until all piles of two or one object left . The player who plays last is a winner. Work out the winning strategy on it.

## Decide which player has winning strategy in maximum matching problem

Given the following game: Two players, player 1 and player 2, play a game in which the first player starts naming a hero $$h_1$$, then player 2 responds with a villain $$v_1$$ who has played in the same movie as $$h_1$$. Then player 1 responds with another hero $$h_2$$ who has played in the same movie as $$v_1$$, and so forth. Each hero and villain can only be used once. The first player that gets stuck, has no more hero/villain, loses the game. Note that player 1 always starts.

The two players may only pick heroes and villains from given sets of heroes $$H$$ and villains $$V$$ ($$|H| = |V| \geq 1$$). They also get a set of movies $$M$$ with the corresponding heroes and villains appearing in that movie.

The question is: can you, based on $$H$$, $$V$$ and $$M$$, decide which player has the winning strategy?

Example:

Given the following data: the heroes are Iron Man, Captain America, Thor and Spider-Man. The villains are Whiplash, Ultron, Thanos and Vulture. The movies are Avengers: Infinity War (stars Iron Man, Captain America, Thor, Thanos and Spider-Man) and Spider-Man: Homecoming (stars Iron Man, Vulture and Spider-Man). Can you decide which player has the winning strategy?

My approach is to use maximum bipartite matching to find out which player has the winning strategy, because we can split the data in two sets, namely $$H$$ and $$V$$ and have relations between those two sets. The Hopcroft-Karp algorithm can take two of such sets and find out the maximum cardinality. Please correct me if I’m wrong: in the cases in which there is a perfect matching, player 2 wins and otherwise player 1 wins. Whenever there is a perfect matching, it means that player 2 has always had an answers to the hero that player 1 named.

How would you solve this? Is there a better, more efficient solution than some maximum bipartite matching.

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## Conceding conflicts and what happens after and how much should the winning party gain?

I am starting a game based somewhat off Atomic Robo the Roleplaying Game which is based off Fate Core and in this case it should not really matter.

This will be my first time GMing and first time playing a Fate based game for all involved, so I am trying to come up with some easy to drop in situations to teach some mechanics through “show, don’t tell”(I will tell too) and to make sure I have a good grasp of these mechanics.

Main (general) questions: When conceding a conflict how much control over the narration does the conceding party have(as long as it is a clear failure of their goals)? Can, or maybe should, conceding party be pushed into other conflict mechanics if it makes narrative sense?

Below is a concrete situation I have been thinking about which brought up these questions: Say I have some NPCs in a cult which are trying to steal tech from Tesladyne for some nefarious purpose. Well they start invading, and the PCs and ally NPCs want to stop this invasion and probably want to figure out what they are trying to do and why. The PCs are winning and so the final couple of enemies NPCs concede the conflict.

What I want to do as a GM is have them escape. So in this case the PCs get some of what they want(stopping the invasion), the enemy didn’t get any of what they wanted. Perhaps clues are left behind so that the PCs can, with some work, figure out who these people are. So far so good, seems like it is in the spirit of a concede.

Say the PCs are not happy with this concede and want the enemy NPCs captured, and this would basically give the PCs everything they wanted in the conflict and that basically defeats the purpose of the concede. I see the situation happening a couple of ways, although not sure if I am totally happy with either:

• I say “well this is a concede I get to narrate what happens as long as it is it isn’t undermining the victory.” Perhaps I narrate smoke bombs or a beam falling down giving the enemy NPCs a chance to escape(should I use a fate point from the reserve to declare details like this for my NPCs?).

• Or I could say “OK lets enter a contest and give you all a chance to catch them before they escape.” This one feels somewhat better to me, but I also am not sure how I feel about someone who concedes pushed into some other conflict mechanic. If the shoe was on the other foot and the PCs concede, would this make sense and should I do it? I do think it fits into the “make things dramatic” scenario, but the purpose of the concede was to get out of conflict(and not let them be able to take these members in).

US politics.

## How many skill points should I give my players for winning an encounter?

I am about to start my first game with The Dark Eye system, and I am wondering how many skill points I should give my players for winning an encounter (not necessarily combat).

Specifically, I am asking for experiences other GMs have made with The Dark Eye and for a rough estimate that I should aim for when rewarding my players. The rulebook only states that a “small adventure” should offer about 5-10 and an “adventure that spans over 2-3 evenings” should be about 15-25 points. However, I like to give rewards on a per-encounter and not on a per-adventure basis, so these estimates aren’t really worth much to me.

How many skill points should I give my players for winning an encounter?

## How many skill points should I give my players for winning an encounter?

I am about to start my first game with The Dark Eye system, and I am wondering how many skill points I should give my players for winning an encounter (not necessarily combat).

Specifically, I am asking for experiences other GMs have made with The Dark Eye and for a rough estimate that I should aim for when rewarding my players. The rulebook only states that a “small adventure” should offer about 5-10 and an “adventure that spans over 2-3 evenings” should be about 15-25 points. However, I like to give rewards on a per-encounter and not on a per-adventure basis, so these estimates aren’t really worth much to me.

How many skill points should I give my players for winning an encounter?

## Winning strategy using Dynamic Programing

Let $$G=(V,E)$$ be a DAG and let $$v_0\in V$$. Alice and Bob are playing a game in which every player has his own turn and Bob is starting. In every turn $$i$$, the player is picking an edge $$e=(v_i,x)$$, then $$x$$ become $$v_{i+1}$$ and then it becomes the other player’s turn. The player which doesn’t have an edge to choose (the vertex $$v_i$$ has no out edges) is losing.

Propose an algorithm which determines if Bob has a winning strategy.